DISCONTINUOUS LONG-FIBER REINFORCED COMPOSITE PROCESSING ANDFINAL PART STIFFNESS PREDICTIONS

DRAFT

Cong Zhang1 and David A. Jack2

Department of Mechanical Engineering, Baylor University. Waco, TX 76798

Abstract

This work employs the rod chain model of Wang et al. (2006) to study the motion of discreteflexible fibers within a polymer suspension. Model results are presented for individual fibers tostudy variations in the transient effects due to a shearing flow between rigid and flexible fibersystems. The presented results demonstrate that the observed fiber motion period decreases asthe fiber flexure increases, and the results provide insight into the modifications required for theequation of motion of the orientation distribution function for flexible suspensions as well as currentlimitations from existing short fiber theories. A methodology for predicting the material propertiesof a discontinuous long fiber reinforced composite is presented by combining the micro-mechanicalapproach of Tandon and Weng (1984) for the underlying unidirectional stiffness tensor predictionswith the flexible laminate theory of Hsiao and Daniel (1996) along with an adapted version ofthe stiffness homogenization approach discussed by Jack and Smith (2008) for an orientationallyvarying distribution of fibers. The influence of the fiber waviness on the processed compositeproperties is shown from the simulation results.

Introduction

There is considerable industrial demand for lightweight, durable composites composed of dis-continuous long fibers due to their high strength to weight ratios. Existing short fiber models forpredicting the processed composites’ underlying microstructure are quite proficient at capturingthe orientation effects during the processing of the polymer/fiber melt, but these models are onlyvalid for systems where fibers remain straight during processing. It is of an industrial benefit tohave similar models available for long fiber systems to aid in the design process.

The orientation of individual fibers within a processed discontinuous fiber reinforced compositeplays a significant role in defining the bulk properties. During processing, the spatially varyingfiber orientation alters the effective viscous stress of the polymer melt, which directly affects theprocessing parameters thus determining the cost and efficiency of processing [1–4]. Jeffery’sequation [5] has been the foundation for research of the rigid fiber orientation within a suspensionfor decades. Fiber-fiber interactions in semi-dilute and concentrated suspensions of rigid fiberscan also be modeled using a fiber probability orientation distribution approach, and they are oftenassumed to be well represented by the addition of a diffusion term to Jeffery’s equation [6–8].

To obtain further advances in the resulting processed part’s performance, long fibers (oftenloosely, but incorrectly, classified as those with an aspect ratio larger than 20) are desired formanufacturing fiber reinforced composites. A perceived drawback is their tendency to flex duringprocessing [9], which will reduce the anticipated material improvements and in the present contextthis will bring into question the accuracy of the existing models for simulating the motion of shortrigid fibers. Additionally, the bulk orientation and flexure of the fibers during processing will causea significant increase in the apparent viscosity of the fluid, thus inducing a higher shear stresson the fibers [3, 10]. Several models for simulating the motion of long fibers have suggesteddifferent mathematical representations for a flexible fiber, such as a series of beads [2, 11], rods[12] and spheroids/needles [1, 13–16]. In the present paper we employ the rod chain model ofSwitzer [17] as expressed by Wang et al. [12] due to its physical clarity and simplicity in the model

1Current address, Department of Mechanical Engineering, the Ohio State University2Author to who correspondence should be addressed: david jack@baylor.edu

Figure 1: The rod chain model of for a flexible fiber depicting the geometry and the force diagram.

implementation. This simplicity may lend itself in the future to an orientation distribution approachas would be required for practical industrial simulations.

Tucker and Liang [18] studied several micromechanical theories for predicting the elastic prop-erties of unidirectional short fiber reinforced composites, and suggested that the approach of Tan-don and Weng [19] or that of Lielens et al. [20] are the most appropriate of the existing modelapproaches. Advani and Tucker [21] averaged the elastic constants of unidirectional fibers toestimate the elastic properties of a short fiber composite with any given fiber orientation distribu-tion. The closed form solutions for both the stiffness expectation and variance from a known fiberorientation distribution were derived by Jack and Smith [22] using the spherical harmonic basis,and it was demonstrated by Gusev et al. [23] that the orientation averaging method yields reliablestiffness predictions for short-fiber composites in engineering design applications.

Hsiao and Daniel [24] assume the configuration of a deformed flexible fiber to be a planarsinusoidal curve, where the waviness of the fiber is defined as a quantity related to the amplitudeand the range of the curve. In the present paper we will modify the unidirectional fiber model ofTandon and Weng [19] using the approach of Hsiao and Daniel for a wavy fiber composite, andhomogenize the solution using a similar approach to that of Advani and Tucker [21] based on theorientation and flexure results from the Wang et al. [12] model for fiber orientation and flexurewithin the polymer melt.

The Rod Chain Model

The rod-chain model introduced by [17] and used by [12] represents fibers as a chain of Nr rigidrods with each rod composed of N beads forming a dilute suspension within a Newtonian fluidwhere Brownian motion is neglected. Each rod has a unit direction vector to indicate the orientationof the rod, whereas the orientation of the fiber is determined by a unit end-to-end vector pend−to−endas shown in Figure 1.

In the following discussion, we non-dimensionalize quantities through the shear rate γ̇, fiberradius a, and the fluid viscosity η as (see e.g., [12] and [25])

t̂ = tγ̇ l̂ = l/a F̂ =F

πηa2γ̇T̂ =

T

πηa3γ̇(1)

where in Equation (1) the hat symbolˆis used to indicate a dimensionless’ parameter.The changing motion of the fluid surrounding the fiber depicted in Figure 1 causes hydrody-

namic forces and torques to be exerted on the fiber. Assuming the hydrodynamic friction forceexerted on each bead is proportional to the velocity difference between the bead and the flow, thehydrodynamic friction force on rod i is equal to the sum of the forces and torques on each beadwithin a rod given by (see e.g., [12])

F̂hi = 6N(κ̂ · r̂i − v̂i), T̂hi = −2N3(ω̂i − ω̂∞), i = 1 ∼ Nr (2)

2

where N is the number of beads in the rod, Nr is the number of rods in a fiber, r̂i is the centerposition vector of rod i, v̂i is the velocity of the center of rod i, ω̂i is the angular velocity vector ofthe center of rod i, and ω̂∞ is the angular velocity vector of the flow at the center of rod i.

Similarly, the bending torque between two neighboring rods is proportional to the differencebetween the bending angle and the equilibrium angle given by (see e.g., [12])

T̂bi = −k̂b(θbi − θbeq)nbi , i = 2 ∼ Nr (3)

where nbi =pi−1×pi

∥pi−1×pi∥ is the unit vector normal to the plane of bending of rod i, the bending angle

is θbi = cos−1

(pi−1·pi

∥pi−1·pi∥

)and the bending torques at the ends of the fiber are each zero. In the

present study we assume all fibers are straight when there are no externally applied loads, thusθbeq = 0. k̂b =

kb

πηa3γ̇where the bending constant kb is related to the bending stiffness of the fiber

via kb = EIb

2Na , where Ib = πa

4

4 .The twisting torque between two neighboring rods is expressed as (see e.g., [12])

T̂ti = −k̂t(θti − θteq)pi, i = 2 ∼ Nr (4)

where a body-fixed unit vector ui perpendicular to rod i is defined in order to determine the twistingangle θti = cos

−1(

ui−1·u′i∥ui−1·u′i∥

), where u′i = ui−1 − (ui−1 · pi)pi, and the twisting torques at the ends

of the fiber are zero. k̂t = kt

πηa3γ̇where the torsion constant kt is related to the torsion stiffness

of the fiber via kt = GIt

2a , where It = πa

4

2 . In the present study fibers are assumed to be initiallystraight, thus the equilibrium twist angle is θteq = 0.

The force equilibrium equations yield the translation equations for each rod as (see e.g., [12])

dv̂i

dt̂=

1

2NRe

(F̂hi + X̂i − X̂i+1

), i = 1 ∼ Nr (5)

where X̂i is the internal constraint force between rod i and i + 1, Re = πa2ργ̇η is the particle

Reynolds number, and ρ is the density of the fiber. Similarly, the moment equilibrium equationsyield the angular equation of motion for each rod as [12]

dω̂i

dt̂=

1

2N3Re[T̂hi + T̂

bi − T̂bi+1 + T̂ti − T̂ti+1 −Npi × (X̂i + X̂i+1)], i = 1 ∼ Nr

The velocities and displacements at the joint of two neighboring rods are the same by continuity,and using the approach shown by the authors in Zhang [25] the internal constraint forces betweenneighboring rods can be explicitly solved for. Once the internal forces are known X̂i the positionand direction vector of each rod can be obtained using the algorithm proposed in Zhang [25].

In this research, we employ the in-house software written by the authors in [25] for the fiber mo-tion and flexure. We employ a fourth-order Runge-Kutta solver to solve the equations of translationand rotation given in Equation (5) and (6) along with a high speed matrix solver for mid-scale sys-tems of equations (dimension ≤ 1, 000) to solve for the internal constraint forces. This approachgreatly improves on the accuracy of the finite difference approach in [12] while reducing the totalnumber of function evaluations by reducing the total number of steps for a desired precision asthe Runge-Kutta solver is of error O

(∆t4

)whereas the finite difference approach is of the error

O (∆t), where ∆t is the time step.

Stiffness of Flexible, Unidirectional Fiber Reinforced Composites

3

z

x

y

z’

x’

y’

pend-to-end

(θ,φ,β)

φ

θ β

(a) (b)

Figure 2: Flexible fiber orientation and configuration (a) Global coordinate description and (b) localcoordinate system along with sinusoidal fit.

The micromechanical model used to predict the elastic properties of a unidirectional compositewith uniform fiber waviness is based on a model proposed by Hsiao and Daniel [24]. The con-figuration for a discontinuous long fiber is assumed to be well described by a planar sinusoidalcurve in some local coordinate frame as depicted in Figure 2. The fiber waviness α is defined asthe amplitude of the fiber in the local coordinate system α = 2πAL where A and L are the am-plitude and the range of the sinusoidal curve, respectively. This result can be approximated in aleast-squares sense from the rod-chain fiber equations of motion as a function of time from thenumerical solutions of Equations (5) and (6).

Hsiao and Daniel divide the volume into infinitesimally thin slices and obtain the compliance ofeach slice from the compliance transformation relations for their unidirectional infinite aspect ratiofibers. The strains are then integrated over each slice to yield the elastic properties of the com-posite. In this study we assume the unidirectional straight fiber compliances used in the Hsiao andDaniel integrations are calculated based the Tandon-Weng theory [19] and as will be discussed inthe next section are valid for small homogeneous regions. As discussed in Zhang [25], the influ-ence of the fiber aspect ratio and waviness will come into play more as the ratio Ef/Em becomeslarger. Zhang demonstrated that the balance between the aspect ratio of the fibers and the wavi-ness determines the final properties of the material and that the desired enhanced properties forlong fiber reinforced composites will be undermined by the increased propensity for fibers to flexdue to processing effects.

Orientation HomogenizationThe material stiffness tensor for a orientationally varying distribution of fibers may be addressedthrough the orientation homogenization approach discussed by the author in Jack and Smith [22].We modify our previous approach for straight short-fibers to extend the method to long wavy dis-continuous fiber composites as depicted in Figure 3. The first step is to decompose the whole partinto a set of smaller aggregates, the orientation of each of which is represented by the orientationof a flexible fiber. Thus each aggregate is treated as a single unidirectional wavy fiber reinforcedcomposite with the same concentration of fibers as the representative volume, of which the localstiffness tensor is predicted as discussed in the preceding section. Next the expectation of thestiffness from the aggregates of the unidirectional wavy fibers is computed assuming a locallyhomogeneous fiber orientation distribution function. Unlike rigid fibers this orientation distributionfunction will have a third degree of freedom as there is no longer symmetry along the fiber direc-tion vector due to flexure. The three angle parameters are shown in Figure 2 where θ is the anglebetween the fiber end-to-end vector and the global z-axis, ϕ is the angle between the projection ofthe end-to-end vector on the xy-plane and the global x-axis, and β is the angle between the fiberplane and the plane formed by the global z-axis and the projection of the end-to-end vector on thexy-plane.

The expectation value of the material stiffness tensor after the composite is processed may beformulated from the non-correlated aggregate of unidirectional fibers defined as the first moment of

4

First

HomogenizationSecond

Homogenization

Figure 3: Orientation homogenization model.

the fiber orientation probability distribution function about the underlying micromechanical stiffnesstensor as a closed form integral expression [22]. In the present context a closed form of the fiberorientation distribution function is not available as only a finite number of fiber samples can beobtained. Thus given a set of N angle combinations {(θn, ϕn, βn) : n = 1, 2, 3, ..., N} correspondingto each of the N unidirectional waviness fiber aggregates, the sample mean Cijkl for the stiffnesstensor from the corresponding stress field is

Cijkl =1

N

N∑n=1

(QnqiQnrjQ

nskQ

ntlC̄qrst) (6)

where it is assumed that the mean strain is constant over each aggregate. Qnij ≡ Qij(θn, ϕn, βn)is the rotation matrix for the coordinate system defined in Figure 2, and C̄pqrs is the unidirectionalwavy fiber stiffness tensor of the aggregate with respect to the coordinate system along the ag-gregate’s principal directions. This approach assumes when the processing of the composite isfinished, the part is cured instantly and all fibers maintain their orientations and curvatures.

Simulation Results of Long Fiber Motion

The early research on fiber motion and orientation was performed by Jeffery [5]. His work de-scribes the motion of a single rigid fiber within a Newtonian flow and the Jeffery equation is thebasis of all short fiber models. In the following studies we will compare the flexible fiber solutionsto those predicted from the rigid-fiber model as a function of increased flexibility of a fiber. Theflexibility of a fiber depends on its aspect ratio and stiffness as well as the strength of the flow asdemonstrated by Forgacs et al. [26,27] as

ηγ̇

E=

ln(2rc)− 1.752r4c

(7)

where rc is the critical buckle aspect ratio of a fiber, above which the axial force along the fiber willbe large enough to bend the fiber.

Single Fiber MotionIn the first example a single fiber is placed within a simple shearing flow common to industrialapplications with the velocity vector components expressed as v1 = γ̇x3 and v2 = v3 = 0 wherethe scalar γ̇ is the shear rate. This particular flow field will tend to rotate a fiber about the x2 axis,where it will spend the majority of its time near the x1 axis. This particular flow is selected due tothe periodic nature of the solution which will highlight flaws in solution approaches for the coupledODEs. The first and third components of the end-to-end unit vector p from Figure 1 are shown

5

Figure 4: Motion period (a) (N = 1, Nr = 10, ar = 10, E/ηγ̇ = 2 × 105, rc = 21.13), initial unitdirection vector p = [1, 0, 0]T , (b) (N = 1, Nr = 30, ar = 30, E/ηγ̇ = 2 × 103, rc = 4.75), initial unitdirection vector p = [1, 0, 0]T ).

in Figure 4 for a fiber aspect ratio that is below the critical aspect ratio and one that is above thecritical aspect ratio. The flexible fiber model and the Jeffery model for rigid fibers are in directagreement when the fiber aspect ratio is below that of the critical buckle aspect ratio. Conversely,as the aspect ratio of the fiber increases beyond that of the critical aspect ratio the motion perioddrastically changes and the end-to-end vector no longer follows the Jeffery path.

Impact of Fiber Flexibility on Fiber MotionWe next study the sensitivity of the period of the fiber’s motion as a function of increasing stiffnessover the range of values for the ratio Eηγ . The values selected,

Eηγ ∈

{104, 2× 105

}, encompass a

range of critical buckle aspect ratios of rc ∈ {8.6, 21.1}. We choose to study a fiber with a geometricaspect ratio of 30, which corresponds to an equivalent fiber ellipsoidal aspect ratio of ∼ 22 (seee.g., [28] for the equivalence calculation). The flexibility range is chosen to highlight a fiber thatis relatively stiff (i.e., when rc ' ar) with that of a flexible fiber (i.e., when rc < ar). We select aninitial orientation of (ϕ, θ) = (0.07, 0.80) subject to the same shearing flow as the previous examplewith flow in the x1 direction and shear in the x3 direction. Results are tabulated for the end to endvector of each of the various fibers, and the results for the x1 component for each of the fibers arepresented in Figure 5. As observed in Figure 5(a) it can be seen that for the first 125 seconds offlow time there is very little difference in the fiber motion for the entire range of fiber flexibilities,but at the onset of fiber tumbling the motion path is a function of fiber flexibility where the fiberbegins to tumble at earlier moments in time as the fiber stiffness decreases. For the stiffer fibersthe x1 component of fiber end-to-end vector goes from positive to negative as one would expectedfrom the Jeffery orbit. This observation impacts the industrial implementation of long fibers in thatfor a macroscopic part composed of many fibers, the reduction of the Jeffery period will causethe peak of the largest component of the second-order orientation tensor (a net approximation ofmicrostructure alignment, see e.g. [21]), will be reduced thus leading to a reduction in the overallpart stiffness, regardless of whether the waviness remains during part curing.

What is interesting to note with the end-to-end vector plots in Figure 5 is that for the flexiblefibers with aspect ratios near the critical buckle aspect ratio they follow a typical Jeffery path wherethe fiber spends as much time pointing along the +x1 axis as it does the −x1 axis. Conversely, asthe fiber flexibility increases this Jeffery characteristic is no longer true as can be seen in Figure5(b) which focuses on the ranges of time where fiber tumble occurs. Observe that the highlyflexible fibers (indicated by the lines with the blue color) apparently flip twice in rapid succession.This characteristic is never seen in rigid fibers, but this is caused by the extreme bend of the fiberwhere it is effectively bent upon itself. For the highly flexible fibers as they are bent during the flow,the trailing end will be pulled around in rapid fashion causing it to unfold.

The waviness of the fibers is presented in Figure 6(a) as a function of time for the same fiber

6

0 50 100 150-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Unitless Time (γ t)

p1

E = 0.11 x 105

γ ηE = 0.13 x 10

5 γ η

E = 0.15 x 105

γ ηE = 0.18 x 10

5 γ η

E = 0.22 x 105

γ ηE = 0.26 x 10

5 γ η

E = 0.30 x 105

γ ηE = 0.36 x 10

5 γ η

E = 0.43 x 105

γ ηE = 0.51 x 10

5 γ η

E = 0.60 x 105

γ ηE = 0.72 x 10

5 γ η

E = 0.85 x 105

γ ηE = 1.01 x 10

5 γ η

E = 1.20 x 105

γ ηE = 1.42 x 10

5 γ η

E = 1.69 x 105

γ ηE = 2.00 x 10

5 γ η

(a) (b) 110 115 120 125 130 135 140 145-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Unitless Time ( γ τ)

p1

E = 0.11 x 105

γ η

E = 0.13 x 105

γ η

E = 0.15 x 105

γ η

E = 0.18 x 105

γ η

E = 0.22 x 105

γ η

E = 0.26 x 105

γ η

E = 0.30 x 105

γ η

E = 0.36 x 105

γ η

E = 0.43 x 105

γ η

E = 0.51 x 105

γ η

E = 0.60 x 105

γ η

E = 0.72 x 105

γ η

E = 0.85 x 105

γ η

E = 1.01 x 105

γ η

E = 1.20 x 105

γ η

E = 1.42 x 105

γ η

E = 1.69 x 105

γ η

E = 2.00 x 105

γ η

Figure 5: End-to-end unit vector component in x1 direction over range of fiber flexibilities.

0 50 100 1500

0.5

1

1.5

2

2.5

3

3.5

Unitless Time (γ t)

Wav

ines

s a

E = 0.11 x 105 γ ηE = 0.13 x 105 γ ηE = 0.15 x 105 γ ηE = 0.18 x 105 γ ηE = 0.22 x 105 γ ηE = 0.26 x 105 γ ηE = 0.30 x 105 γ ηE = 0.36 x 105 γ ηE = 0.43 x 105 γ ηE = 0.51 x 105 γ ηE = 0.60 x 105 γ ηE = 0.72 x 105 γ ηE = 0.85 x 105 γ ηE = 1.01 x 105 γ ηE = 1.20 x 105 γ ηE = 1.42 x 105 γ ηE = 1.69 x 105 γ ηE = 2.00 x 105 γ η

110 115 120 125 130 135 140 145-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Unitless Time (γ t)

Wav

ines

s α

E = 0.26 x 105 γ ηE = 0.30 x 105 γ ηE = 0.36 x 105 γ ηE = 0.43 x 105 γ ηE = 0.51 x 105 γ ηE = 0.60 x 105 γ ηE = 0.72 x 105 γ ηE = 0.85 x 105 γ ηE = 1.01 x 105 γ ηE = 1.20 x 105 γ ηE = 1.42 x 105 γ ηE = 1.69 x 105 γ ηE = 2.00 x 105 γ η

(a) (b)

Figure 6: Fiber waviness during shearing flow for a range of fiber flexibilities.

motions shown in Figure 5. For the most flexible fibers (again, those lines indicated by the bluecolor) the waviness ratio is as large as 3. This large waviness makes these fibers quite ineffectivefor composite products as the part stiffness will be greatly reduced. Conversely, for the fibers whichfollow the Jeffery period behavior, albeit with a visible shift in orbit behavior, the fiber waviness isnot unreasonably large as observed in Figure 6(b) where just the fibers with the Jeffery-like orbitalpaths are retained. It is clear that the peak of the waviness function occurs around the same timeas when the fiber tumble occurs with the trend that the lower the relative stiffness of the fiber thelarger the waviness peak becomes. It is somewhat troubling from a manufacturing perspectiveto see the relatively large value of the waviness ratio for these remaining fibers as it was shownin Zhang [25] that a waviness value of 0.2 ∼ 0.4 can reduce the effective stiffness by 50% for acomposite where the stiffness of the fiber is 100 times that of the matrix as might be expectedfrom a chopped carbon fiber polymeric composite. To reduce these peaks, one would have toincrease the relative fiber stiffness, which can be done by either increasing the modulus of thefiber or reducing the injection rate, ηγ, of the flow, which can be accomplished by either increasingthe melt temperature or reducing the mold inlet flow rate.

Simulation Results of Prediction of Material Stiffness

We sampled 1,000 fibers from an initially isotropic fiber orientation distribution, and solved theequation of motion from Equations (5) and (6) for each fiber for the same flow scenario as in theprevious examples. The stiffness of the fibers relative to the flow is E/ηγ = 2×104, correspondingto a critical buckle aspect ratio of 10.7, which is much smaller than the geometric aspect ratio of thefibers 30, and the Reynolds number of the flow is Re = 0.1 with a shear rate of γ = 1.0sec−1. Theratio of fiber to matrix stiffness is Ef/Em = 100, with a fiber volume fraction of vf = 3% and aspect

7

Figure 7: The predicted material properties of a processed part (Ef/Em = 100, vf = 0.03, νf =0.30, νm = 0.35, ar = 30, Nf = 1000): (a) longitudinal modulus E11; (b) transverse modulus E22;and (c) major Possion’s ratio ν12.

ratio of ar = 30, and a fiber and matrix Poisson Ratio of, respectively, νf = 0.30 and νm = 0.35.Taking the homogenization approach suggested in Equation (6) we compute the expectation of thestiffness tensor and present the results in Figure 7 for predicted values of the longitudinal modulusfor a composite part.

Notice that the part stiffness shown in Figure 7 goes through a similar periodic behavior as wasobserved in the orientation of a single fiber, the difference is these results are for a distribution offibers within a flow, each with a different initial orientation as would be expected from an injectionmolded part. This periodic behavior is not smooth nor repeating until after several complete cycles.This nature is quite different than the motion that would be predicted by the Jeffery model forrigid fibers. The Jeffery model predicts that the motion is periodic with a smooth and repeatableorientation pattern.

There are two curves provided in each of the plots of Figure 7. Both curves are based onthe orientation results from the flexible fiber motion study, but the blue curve corresponds to thematerial properties computed using a straight fiber micro-mechanics model and the green line cor-responds to the material properties computed from the flexible fiber model discussed previously.The discrepancy between the respective predicted longitudinal moduli E11 under the two circum-stances is due to the flexure of the fibers. To be more specific, without considering the wavinessof the fibers in the fiber stiffness predictions, an overly optimistic longitudinal modulus E11 will bepredicted where there are large fiber waviness values. It is unclear as to whether the fibers willstraighten out once the flow stops and prior to the hardening of the polymeric matrix. We expectthat the true solution lies somewhere between the two lines as we expect some un-flexing oncethe mold filling ceases prior to the hardening of the matrix.

Conclusions

A methodology has been discussed to predict the fiber orientation and flexure of a long-fiber injec-tion molded composite. The results for orientation suggest that short-fiber models may be applica-ble for long-fiber orientation simulations if an appropriate reduction to the bulk average orientationis made. Results are presented to demonstrate at which point the short fiber model approachwould no longer be appropriate and more complicated techniques for bulk average orientationsand waviness are required. We also discuss a technique to predict the material properties of longfiber reinforced composites with fibers having a distribution of orientations in the composites. Theresults presented focus on the scenario where the fiber orientation and flexure was obtained fora highly flexible fiber set, and it was shown that using classical approaches for stiffness predic-tions from the orientation state of fibers will over predict the stiffness if fibers remain bent due toprocessing considerations.

8

Acknowledgements

The authors gratefully acknowledge support from the N.S.F. via grant C.M.M.I. 0727399 as well asthe support from Baylor University for their new faculty start-up package.

References

1. C.G. Joung, N. Phan-Thien, and X.J. Fan. Viscosity of Curved Fibers in Suspension. Journalof Non-Newtonian Fluid Mechanics, 102:1–17, 2002.

2. S. Yamamota and T. Matsuoka. Dynamic Simulation of Microstructure and Rheology of FiberSuspensions. Polymer Engineering Science, 36(19):2396–2403, October 1996.

3. B.E. VerWeyst and Tucker C.L. III. Fiber Suspensions in Complex Geometries: Flow-Orientation Coupling. The Canadian Journal of Chemical Engineering, 80:1093–1106, De-cember 2002.

4. B.E. VerWeyst, C.L. Tucker, P.H. Foss, and J.F. O’Gara. Fiber Orientation in 3-D InjectionMolded Features: Prediction and Experiment. International Polymer Processing, 14:409–420,1999.

5. G.B. Jeffery. The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid. Proceedings ofthe Royal Society of London A, 102:161–179, March 1923.

6. F.P. Folgar and C.L. Tucker. Orientation Behavior of Fibers in Concentrated Suspensions. Jn.of Reinforced Plastics and Composites, 3:98–119, April 1984.

7. J.H. Phelps and C.L. Tucker. An Anisotropic Rotary Diffusion Model for Fiber Orienation inShort- and Long Fiber Thermoplastics. Journal of Non-Newtonian Fluid Mechanics, 156:165–176, 2009.

8. Montgomery-Smith, S.J., D.A. Jack, and D.E. Smith. A Systematic Approach to ObtainingNumerical Solutions of Jefferys Type Equations using Spherical Harmonics. Composites,Part A, 41:827–835, 2010.

9. N. Nguyen, S.K. Bapanapalli, J.D. Holbery, M.T. Smith, V. Kunc, B. Frame, J.H. Phelps, andC.L. Tucker. Fiber Length and orientation in Long-Fiber Injection-Molded Thermoplastics - PartI: Modeling of Microstructure and Elastic Properties. Jn. of Composite Materials, 42(10):1003–1029, 2008.

10. S.M. Dinh and R.C. Armstrong. A Rheological Equation of State for Semiconcentrated FiberSuspensions. Jn. of Rheology, 28(3):207–227, 1984.

11. S. Yamamota and T. Matsuoka. A Method for Dynamic Simulation of Rigid and Flexible Fibersin a Flow Field. Journal of Chemical Physics, 98(1):644–650, 1993.

12. G. Wang, W. Yu, and C. Zhou. Optimization of the Rod Chain Model to Simulate the Motionsof a Long Flexible Fiber in Simple Shear Flows. European Journal of Mechanics, 25:337–347,2006.

13. R.F. Ross and D.J. Klingenberg. Dynamic Simulation of Flexible Fibers Composed of LinkedRigid Bodies. Jn. of Chemical Physics, 106:2949–2960, February 1997.

14. P. Skjetne, R.F. Ross, and D.J. Klingenberg. Simulation of single fiber dynamics. Journal ofChemical Physics, 107(6):2108–2121, 1997.

15. C.F. Schmid, L.H. Switzer, and D.J. Klingenberg. Simulations of fiber flocculation: Effects offiber properties and interfiber friction. Journal of Rheology, 44:781–810, 2000.

9

16. L.H. III Switzer, Leonard H., and D.J. Klingenberg. Rheology of sheared flexible fiber suspen-sions via fiber-level simulations. Journal of Rheology, 47(3):759–778, May 2003.

17. L.H. III Switzer. Simulating Systems of Flexible Fibers. PhD thesis, University of Wisconsin-Madison, December 2002.

18. C.L. Tucker and E. Liang. Stiffness Predictions for Unidirectional Short-Fiber Composites:Review and Evaluation. Composites Science and Technology, 59:655–671, 1999.

19. G.P. Tandon and G.J. Weng. The Effect of Aspect Ratio of Inclusions on the Elastic Propertiesof Unidirectionally Aligned Composites. Polymer Composites, 5(4):327–333, 1984.

20. G. Lielens, P. Pirotte, A. Couniot, F. Dupret, and R. Keunings. Prediction of Thermo-Mechanical Properties for Compression Moulded Composites. Composites Part A, 29A:63–70, 1998.

21. S.G. Advani and C.L. Tucker. The Use of Tensors to Describe and Predict Fiber Orientation inShort Fiber Composites. Jn. of Rheology, 31(8):751–784, 1987.

22. D.A. Jack and D.E. Smith. Elastic Properties of Short-Fiber Polymer Composites, Derivationand Demonstration of Analytical Forms for Expectation and Variance from Orientation Ten-sors. Journal of Composite Materials, 42(3):277–308, 2008.

23. A. Gusev, M. Heggli, H.R. Lusti, and P.J. Hine. Orientation Averaging for Stiffness and Ther-mal Expansion of Short Fiber Composites. Advanced Engineering Materials, 4(12):931–933,2002.

24. H.M. Hsiao and I.M. Daniel. Elastic Properties of Composites with Fiber Waviness. Compos-ites, Part A, 27(10):931–941, 1996.

25. C. Zhang. Modeling of Flexible Fiber Motion and Prediction of Material Properties. Master’sthesis, Baylor University, 2011.

26. O.L. Forgacs and S.G. Mason. Particle motions in sheared suspensions: IX. Spin and defor-mation of threadlike particles. Journal of colloid science, 14(5):457–472, 1959.

27. O. L. Forgacs and S. G. Mason. Particle motions in sheared suspensions : X. orbits of flexiblethreadlike particles. Journal of Colloid Science, 14:473–491, October 1959.

28. D. Zhang, D.E. Smith, D.A. Jack, and S. Montgomery-Smith. Numerical Evaluation of SingleFiber Motion for Short-Fiber-Reinforced Composite Materials Processing. Journal of Manu-facturing Science and Engineering, 133(5):051002–9, 2011.

10