A Theoretical Model of Concurrent Longitudinal andCircumferential Superdrawing of Hollow PolyethyleneTerephthalate Fibers

Youjiang Wang,1 Arun Pal Aneja21 School of Polymer, Textile and Fiber Engineering, Georgia Institute of Technology, Atlanta, Georgia

2 Noeton Policy In Innovation, Greenville, North Carolina

In a superdrawing process, a polyethylene terephtha-late (PET) filament is elongated without developingmuch orientation and crystallization. Exploiting thisphenomenon may bring about lower cost, more flexibleand faster response in synthetic fiber production. Theconcurrent longitudinal and circumferential superdraw-ing phenomenon of PET hollow fibers is explainedusing the viscoelastic behavior of a thick walled cylin-der under an internal pressure and an axial load in acontinuous process. The model defines the stress–strain-displacement relationship of hollow fibers. Thefiber undergoes instantaneous radial superdrawing(increase in thickness) in the process zone followed byconcurrent circumferential (increase in void) and longi-tudinal (increase in length) superdrawing. Based onmaterial viscoelastic properties and processing condi-tions, the model predicts the threadline tension, inter-nal pressure, and final fiber geometries. Excellentagreement of the model with experimental results isobserved over a range of processing conditions. Themodel is developed from a process engineering view-point to enable the analysis of the impact of processparameters during superdrawing on fiber properties.POLYM. ENG. SCI., 50:1773–1779, 2010. ª 2010 Society ofPlastics Engineers

INTRODUCTION

The melt spinning process is the most common and

most economical process to produce synthetic fibers for

textile and technical applications. The process involves

extruding molten polymer through a spinneret. The extru-

date is then stretched and solidified. The filaments are

subsequently drawn to increase molecular alignment and

crystallinity so as to improve their tensile strength and

modulus. Because of the change in molecular orientation

and crystallinity, the fiber draw ratio in the conventional

drawing process is limited to about 3 to 9. There are

some potential benefits if the limit on the draw ratio could

be lifted. Pace [1] discovered that under the condition for

superdrawing, a freshly melt-extruded, amorphous, poly-

ethylene terephthalate (PET) structure, such as an as-spun

yarn, can be drawn up to 75 times its original length

without appreciable orientation or crystallization. The

superdrawing phenomenon is also referred to as flow

drawing, amorphous drawing, or super stretching. As nei-

ther appreciable orientation nor crystallization occurs after

superdrawing, the superdrawn structures have very low

tenacities and moduli (tenacities in the range of 0.4�0.8

gram per denier). To obtain orientation and crystalliza-

tion, the superdrawn structures can be subjected to a con-

ventional drawing to obtain the required tenacities and

moduli.

In contrast to the traditional process, a superdrawing-

based process offers the possibility of using one standard

spun supply from one spinning die to produce fibers with

a wide range of fiber linear densities (deniers) because of

the much wider range of possible fiber draw ratios. This

not only offers a unique method for quick-response manu-

facturing but also a route to produce fine-denier fibers

from as-spun fibers of ‘‘normal’’ textile deniers. The melt-

extrusion phase in fiber melt-spinning represents a very

large percentage of the total cost of production, whereas

the drawing operation is considerably less expensive.

Spinning a large-diameter standard stock allows the ex-

truder to operate at a higher throughput and therefore at

lower cost. Superdrawing-based process has the potential

to be much more efficient, cost effective, and faster than

the traditional approach to meeting the market demand

for fibers with varying finenesses.

As demonstrated by Aneja et al. [2–4], applying super-

drawing to hollow fiber production provides an effective

way to produce hollow fibers with a large hollow core

fraction. Hollow fibers are high value added products used

Correspondence to: Youjiang Wang; e-mail: youjiang.wang@ptfe.

gatech.edu

Contract grant sponsor: United States Department of Commerce,

National Textile Center.

DOI 10.1002/pen.21711

View this article online at wileyonlinelibrary.com.

VVC 2010 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2010

widely for textile, technical, and biomedical applications.

Compared with traditional fibers, hollow fibers offer

reduced effective fiber density and increased effective fiber

surface area. Low fiber density is important to maximize

thermal insulation and high fiber surface area is important

to maximize moisture transport, both features are critical

in thermal wear. Low linear density contributes to espe-

cially soft tactile aesthetics. Traditionally, the production

of hollow fibers involves spinning of the hollow fiber fol-

lowed by a drawing process to reduce its cross sectional

area (linear density). In the spinning process, the polymer

melt and air are extruded simultaneously through the spin-

neret exit, much like the production of bicomponent fibers.

When the drawing takes place in air, the void (hollow

core) of the fiber decreases accordingly. Aneja and Wang

[3] reported the use of steam instead of hot water for

improved operability. Other liquids such as methanol and

ethanol may also be used with even better effectiveness

[4]. During such process water permeates inside the hollow

core and expands as it vaporizes, and the air trapped inside

the void is heated and expanded. Both of these mecha-

nisms exert an outward force which causes circumferential

superdrawing such that the fiber void is increased.

Some theoretical and experimental studies related to

PET superdrawing have been reported in the literature.

Comparing the viscoelastic behavior of PET in water and

in glycerin, Kawaguchi [5] notes that water lowers the

softening temperature of PET by about 208C. This seems

to confirm that superdrawing can occur at a much lower

temperature in water than in air or oil. Thompson [6]

demonstrates that drawing of as-spun PET filament with

or without crystallization can be realized simply by alter-

ing how the process is started.

Kawaguchi [5] presented a viscoelastic model of PET

superdrawing, relating the fiber stress as a function exten-

sion as follows, with good agreement with experimental

results:

Fiber stress ¼ E0

1þ attað Þ2 1� e�t=t

h i2þ tað Þ 1� e�t=t

h i� �

(1)

where E0 is initial modulus, a is rate of extension, t is time

constant, and t is time. In the experiment part, Kawaguchi

[5] measured the stress–strain curve of PET under constant

rates of extension at different temperatures. He also calcu-

lated the viscosity coefficient versus temperature. He found

that the viscosity coefficient depends not only on the tem-

perature but also on the rate of extension.

Studying the circumferential superdrawing of hollow

PET in water, Aneja [2] derived an equation to predict

the hollow fiber void content as a function of time (t)from initial geometry, temperature, and water-polymer

interaction (constants A, B, and s):

Voidð%Þ ¼ Aþ Bt 1� e�t=t� �

(2)

The equation agrees well with experimental data, dem-

onstrating that hot water bath can provide internal pres-

sure needed for circumferential superdrawing of hollow

fibers.

Chandran and Jabarin [7] carried out biaxial stretching

experiments on PET films and investigated factors that

influenced the stress–strain curve in the x direction. They

found that the strain in the y direction has a significant

influence on the stretching properties in the x direction:

Increasing the y-direction strain is equivalent to lowering

the sample temperature or increasing the rate of extension

in the x direction. This indicates that the viscoelasticity in

one direction is dependant on the mechanical property

and deformation in other directions. Findley et al. [8]

reviewed multiaxial deformations of plastics, including si-

multaneous creep under one component of stress and

relaxation under a different component of stress. Several

viscoelastic constitutive models are summarized by Find-

ley et al. [9]. Water permeation in polymers has been

discussed by Yasuda et al. [10]. They used a modified

time-lag procedure to determine the diffusion constant

and permeability of water as a function of temperature

through various polymer films.

There is a need for theoretical analysis concerning con-

current longitudinal and circumferential superdrawing of

hollow fibers. This article presents a comprehensive fun-

damental study to understand the various phenomena that

occur in a continuous superdrawing process and their

impact on fiber properties in the manufacture of hollow

PET fibers. A theoretical model is developed for the con-

current longitudinal and circumferential superdrawing

phenomenon. This leads to a better understanding of the

basic processes for improved design, process control,

prediction, optimization, and uniform product quality in

water and steam media superdrawing.

THEORETICAL MODEL OF CONCURRENTSUPERDRAWING

Modeling Procedure

To model the concurrent longitudinal and circumferen-

tial superdrawing process, the fiber inside a drawing zone

is considered. Figure 1 is a schematic used to define the

basic quantities: The fiber is subjected to a constant axial

load (thread tension) and a uniform inner pressure p. Thisinner pressure, giving rise to circumferential superdraw-

ing, is created because of expansion of air inside the void

by the environmental temperature change and vapor pres-

sure because of water permeation from outside to the

void. The fiber’s inner and outer radius are a(0) and b(0),respectively, as the fiber enters the drawing zone at a

speed of V1, and its inner and outer radius become a(L)and b(L), respectively, as the fiber exits the drawing zone

at a speed of V2.

1774 POLYMER ENGINEERING AND SCIENCE—-2010 DOI 10.1002/pen

The polymer material is modeled as a viscoelastic ma-

terial under biaxial tension. Although the axial force and

internal pressure are constant along the fiber, the velocity

and radius profiles are to be determined by the equations,

and therefore the stresses as a function of location (x, r)are unknowns to be solved.

An infinitesimal segment of the hollow fiber at position

x is first analyzed. As indicated in Fig. 2, the hollow fiber

has an inner radius a and an out radius b. To predict the

change in fiber geometry after going through the super-

drawing process, it is necessary to find the elastic solution

for the distribution of stresses, strains, and displacements

in response to combined internal loading p and axial

stress r0. The generalized correspondence principle is

then used to obtain the viscoelastic behavior. As the fiber

cross-sectional area varies along the fiber, the axial stress

varies as well along the fiber length, while the axial force

(load) remains constant. A fiber segment’s residence time

inside the superdrawing machine also increases as it

passes through the drafting zone. To account for the

changes in axial stress and time along the hollow fiber,

Boltzmann’s superposition principle is used.

The general procedure involves the following steps:

(1) to obtain a linear elastic solution to the problem of

a thick-walled tube under biaxial stresses, (2) to derive

a set of governing equations by substituting generalized

material constants (time dependent modulus and

Poisson’s ratio) in the equations obtained in (1), and,

(3) to apply the governing equations to the entire fiber

length in the drawing zone, satisfying the boundary

conditions.

Linear Elastic Solution of Infinitesimal Fiber Segmentunder Axial Stress and Internal Pressure

For the infinitesimal segment of the hollow fiber at x(see Fig. 2), the change of diameters over this short

length is neglected. The linear elastic solution to the prob-

lem of a thick-walled tube (inner radius ¼ a, outer radius¼ b) under simultaneous internal pressure (p) and axial

stress (r0) in the cylindrical coordinate system (r, y, z)for e (strains),ur (radial displacement), and r (stresses) is

as follows:

srrðrÞ ¼ p

ðb=aÞ2 � 11� ðb=rÞ2h i

syyðrÞ ¼ p

ðb=aÞ2 � 11þ ðb=rÞ2h i

szzðrÞ ¼ s0 ¼ F

pðb2 � a2Þezz ¼ s0

E0

� v0E0

2p

ðb=aÞ2 � 1

urðrÞ ¼ pr

ðb=aÞ2 � 1

1þ v0E0

1� v01þ v0

þ b

r

8>: 9>;2� �� rv0s0

E0

(3)

where v0 and E0 are the Poisson’s ratio and elastic modu-

lus of the material.

The stress and strain tensors can be expressed as the

sum of two components: (1) Deviatoric (shear, distortion),

and (2) Dilatational (volumetric, hydrostatic), as,

sij ¼s11 s12 s13s21 s22 s23s31 s32 s33

�������

�������¼

su 0 0

0 su 0

0 0 su

�������

�������þ

s11 s12 s13

s21 s22 s23

s31 s32 s33

�������

��������where

X3i¼1

sii ¼ 0

eij ¼e11 e12 e13e21 e22 e23e31 e32 e33

�������

�������¼

eu 0 0

0 eu 0

0 0 eu

�������

�������þ

d11 d12 d13

d21 d22 d23

d31 d32 d33

�������

��������where

X3i¼1

dii ¼ 0

ðFull TensorÞ ðDilatationalÞ ðDeviatoricÞ ð4Þ

The constitutive relations of a linear elastic material can

be expressed separately for the dilatational and deviatoric

components, respectively,

su ¼ 3K eu and sij ¼ 2G dij (5)

where G is the shear modulus and K is the bulk modulus,

which are related to the Young’s modulus E and Pois-

son’s ratio n by

E ¼ 9KG

3K þ G; n ¼ 3K � 2G

6K þ 2G(6)

FIG. 1. Geometry and boundary conditions.

FIG. 2. A fiber segment at x.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2010 1775

Viscoelastic Solution of Infinitesimal Fiber Segmentunder Axial Stress and Internal Pressure

For a viscoelastic material under multiaxial stresses,

both the stress and strain are functions of time. The con-

stitutive stress–strain relations can be expressed by the

following generic equations [9]:

P1sijðtÞ ¼ Q1dijðtÞ ðdeviatoricÞ (7)

P2suðtÞ ¼ Q2euðtÞ ðdilatationalÞ (8)

where (P1, Q1, P2, and Q2) are linear differential opera-

tors with respect to time, for example,

P1 ¼ p0 þ p1qqt

þ p2q2

qt2þ � � � (9)

The viscoelastic behavior of a polymer under uniaxial

stress is often described by one of the three simple mod-

els: the Maxwell model consisting a spring and a dashpot

in series, the Kelvin model consisting a spring and a

dashpot in parallel, and a Three Element model consisting

a spring connected with a Kelvin model in series. Based

on the nature of polymer deformation in superdrawing,

the Maxwell model provides appropriate description of

the viscoelastic behavior as the polymer undergoes amor-

phous stretching (see Fig. 3). The stress–strain equation

and the corresponding linear differential operators are as

follows,

sþ ZRs� ¼ Z e

� , Ps ¼ Qe

where P � 1þ ZR

qqt

8>: 9>;; Q � Zqqt

8>: 9>; (10)

When subjected to multiaxial stresses, the viscoelastic

behavior of a polymer is much more complicated. The

approaches outlined in [9] are applied to the solution pro-

cedure for this problem. For a thick-walled viscoelastic

cylinder under biaxial stresses shown in Fig. 2, the ‘‘Gen-

eralized Correspondence Principle’’ can be applied, by

replacing materials constants in the linear-elastic solution

(Eq. 3) with the following time dependent materials con-

stants,

G ¼ 1

2

Q1

P1

; K ¼ 1

3

Q2

P2

(11)

Following Eq. 5 and after applying Eq. 6, the following

time-dependent modulus and Poisson’s ratio are obtained,

E0 tð Þ ¼ 3Q1Q2

P2Q1 þ 2P1Q2

; n0 tð Þ ¼ P1Q2 � P2Q1

P2Q1 þ 2P1Q2

(12)

It is then assumed that the deviatoric deformation of the

polymer under biaxial tension follows the Maxwell Model

of viscoelasticity, and the dilatational deformation of the

polymer under biaxial tension is governed by linear elas-

ticity. Thus, (P1, Q1, P2, and Q2) in Eq. 12 are given by

P1 ¼ 1þ ZR

qqt

8>: 9>;; Q1 ¼ Zqqt

8>: 9>;; and P2 ¼ 1; Q2 ¼ 3K

(13)

where R and Z are material viscoelastic constants, meas-

ured by, for example, a creep, a stress relaxation, or a

tensile test (e.g., [5]), and K can be calculated from the

initial material constants, K ¼ E/3(1 � 2v). Placing Eq.13 in Eq. 12, and then in Eq. 3, the time depend load-dis-

placement-strain relations of a fiber segment modeled as a

thick-walled tube in Fig. 2 are obtained:

ur r; tð Þ ¼ f r; t; p; s0; a; b;R; Zð Þezz tð Þ ¼ f t; p; s0; a; b;R; Zð Þ (14)

from which the new inner and outer radii and length after

time t can be calculated.

Viscoelastic Solution of Entire Fiber Length in aSuperdrawing Zone

In a drawing process depicted in Fig. 1, the thread ten-

sion (related to axial stress r0), internal pressure (p), andresidence time are all unknowns which are determined by

the processing conditions and material properties includ-

ing, (1) initial and final velocities (V1, V2), temperature

(T) and process zone length (L), (2) initial fiber geometry

(a, b), and (3) material properties (R, g).Therefore, the process parameters, such as the final

fiber cross sectional shape and thread tension, cannot

directly be calculated from Eq. 11. Instead, the problem

may be solved iteratively, requiring each infinitesimal

segment of the fiber in the drawing zone to satisfy Eq.11. Fig. 4 shows schematically a piece of fiber beforeFIG. 3. The Maxwell model.

FIG. 4. Schematic of fiber segment before and after entering the

process zone.

1776 POLYMER ENGINEERING AND SCIENCE—-2010 DOI 10.1002/pen

entering the drawing zone, and the same piece of fiber af-

ter it has completely entered the process zone. During the

time period equal to the total residence time tf, the infini-

tesimal fiber segment, dz, enters the drawing zone and

moves to its new position at x, and its new length

becomes dx. As this infinitesimal fiber segment moves

inside the drawing zone, its cross sectional area is reduced

gradually, and this corresponding to a gradual increase in

the axial stress. To account for this change in axial stress

and residence time along the fiber, Boltzmann’s superpo-

sition principle is used. A decrease in cross sectional area

from x to (x þ dx) corresponds to an increase in the axial

stress, Dr0 at position x and time t.Figure 5 illustrates the procedure for obtaining a numeri-

cal solution to the problem illustrated in Fig. 4 based on

the equations developed. Initial estimated values for the

axial stress (r0) at the entrance of the drawing zone (x ¼0) and the internal pressure (p) are used in the calculation

and their values are updated after each step of iteration until

desired accuracy is reached when the calculated exit veloc-

ity V2 is equal to the actual fast roller speed V2, and the

calculated internal pressure p is equal to the p value used

as input for the computational procedure. The total resi-

dence time (tf) is determined such that the length of a fiber

segment going into the drawing zone over this period of

time (length ¼ V1 tf) is equal to the process zone length

(L), by the following equation (see Fig. 4):

xðtf Þ ¼Z V1tf

z¼0

1þ ezz t ¼ z

V1

� � �dz ¼ L (15)

and the internal pressure (p) is determined from

p0T0

U0 ¼ p

TU (16)

where U is the volume of the core of the hollow fiber.

The viscoelastic constants (R, g) for the material at a

given processing temperature are obtained from Ref. 5

and used in the calculations.

EXPERIMENTAL VERIFICATION

Experimental work was conducted on a drawing

machine with a hot water bath. An amorphous PET fiber

supply, spun at 1400–1600 m/min, was used in the experi-

mental study. The processing conditions were: V1 ¼ 10.0

m/min, V2 ¼ 22.9 m/min, and water bath temperature ¼988C. Data was collected to represent the state of the fiber

in the superdrawing bath from entrance to exit. The sam-

ples were obtained by stopping the draw machine, mark-

ing the entrance and exit, and quickly removing the seg-

ment of the fiber in the bath. This segment was immedi-

ately quenched in cold water and then left to dry under

a hood. The fiber geometry was analyzed at fixed inter-

vals of 279.4 mm for the entire length of the sample

(�4.47 m).

Figure 6 compares the experimental results with the

theoretical predictions for the fiber inner and outer diame-

ters and the void ratio along the fiber threadline in the

FIG. 5. Flow chart for the solution procedure.

FIG. 6. Comparison between theoretical predictions and experimental

results for superdrawing in water bath: (a) fiber inner and outer diame-

ters versus position, (b) core void versus position.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2010 1777

superdrawing bath. On entry into the superdrawing bath

the inner and outer diameters are 40 lm and 77 lm,

respectively. Both these dimensions then decline gradually

from these values to 25 lm and 46 lm, respectively, upon

exiting the process zone (Fig. 6a). Figure 6b shows

change in fiber void in the superdrawing bath. The void

increases gradually with distance from 25 to 35% from

start-to-finish. The axial force required to achieve the dis-

placements at these conditions is 1.216 mN.

A neck is often formed when a synthetic fiber is sub-

jected to a conventional drawing process, whereas in a

superdrawing process the reduction in fiber linear density

occurs gradually over the entire drawing zone. The

observed changes in diameters and void indicate that the

drawing in water bath is not a neck-drawing phenomenon

but the drawing takes place over the entire zone, exhibit-

ing a taper-drawing phenomenon.

Overall, a good agreement between calculated results

and experimental data is observed, which confirms the va-

lidity of the theoretical model. The theoretical prediction

for fiber diameters agrees especially well with experimen-

tal results corresponding to the first half in the superdraw-

ing zone, and the theory over estimates the fiber diame-

ters near the end of the process zone. As the model

assumes the material is isotropic, this discrepancy could

be due to the possible development of some molecular

orientation and some anisotropy in material properties,

which results in higher lateral contraction when stretched

axially. Although crystallization must be suppressed in

superdrawing, some low level of molecular orientation

may be present in amorphous PET when the fiber is elon-

gated at temperature above Tg. Effort is underway to char-

acterize the behavior of amorphous orientation during

superdrawing and refine the model to account for such

effect. The predicted fiber void agrees well with experi-

mental results over the entire process zone.

Figure 7 shows the stress distribution profiles on the

inside and outside fiber walls of the hollow fiber during

superdrawing. The rr, sq and rz are three components (ra-

dial, tangential, and axial) of the stresses on the inner sur-

face. Both sq and rz are positive and increasing with dis-

tance in the superdrawing bath, indicating the fiber is under

tension in these two directions. The axial stress rzincreases slowly at first (367 kPa at x ¼ 0) and then rapidly

toward the end (713 kPa at x ¼ 4470 mm), whereas for sqthe change is gradual from 147 kPa to 182 kPa. The radial

stress rr is a compression pressure and remains constant on

the inner surface at �91.7 kPa, corresponding to the vapor

pressure at the processing temperature. There is no radial

stress rr on the outside fiber wall. The axial stress does not

vary with radial position, and is the same on the outside

surface as on the inside surface. The tangential stress sq isa tensile force of low magnitude and increases from 55 to

91 kPa from entrance of the bath to exit.

The strain distribution profile of the inside and outside

fiber walls is shown in Fig. 8. The ez and eq strain compo-

nents represent extension of the fiber in the axial and tan-

gential directions, respectively. The axial strain ezincreases from 0.078 to 0.672 from start-to-finish, while

the tangential strain eq decreases from 0.078 to �0.065.

Thus, we observe that eq goes from a positive quantity

(extension) to a negative one (compression). This is

because over a distance the inside diameter goes below

the initial inlet value due to longitudinal extension, caus-

ing a resultant shrinkage in the circumferential direction.

The strain in the radial direction er represents shrinkage in

the wall thickness direction. The axial strain ez is the

same on the outside fiber surface as on the inside fiber

surface and represents extension. Both er and eq on the

outside fiber wall are negative and represent shrinkage.

These two parameters vary from �0.029 and �0.013 at

the entrance to �0.354 and �0.251 at the exit of the

superdrawing bath, respectively, representing shrinkage in

the radial and tangential directions.

CONCLUSIONS

In a superdrawing process, a polymer filament is elon-

gated without developing much orientation and crystalliza-

FIG. 7. Stress components on the inside and outside surfaces of fiber

along the threadline.

FIG. 8. Strain components on the inside and outside surfaces of fiber

along the threadline.

1778 POLYMER ENGINEERING AND SCIENCE—-2010 DOI 10.1002/pen

tion. Exploiting this phenomenon may bring about lower

cost, more flexible, and faster response in synthetic fiber

production. A theoretical model has been developed to

describe the concurrent longitudinal and circumferential

superdrawing phenomenon of PET hollow fibers in a

dynamic state. Based on material viscoelastic properties

and processing conditions, the model predicts the thread-

line tension, stresses, strains, and final fiber geometries.

Excellent agreement of the model with experimental

results is observed. The model is developed from a process

engineering viewpoint. The impact of process parameter

changes during superdrawing on fiber properties is ana-

lyzed and the results will be reported in a separate paper.

NOMENCLATURE

r, y, z cylindrical coordinates, in the ra-

dial, tangential, and axial direc-

tions

a, b fiber inner and outer radius at

position x

x position of a fiber cross section

from the entrance of the process

zone

z position of a fiber cross section

before entering processing zone

(see Fig. 4)

P internal pressure

V1, V2 fiber velocities at entrance and exit

points, respectively

L processing zone length

t Time

tf total time in the processing zone

(residence time)

U0, U Volume of void before and after

entering the processing zone,

respectively

E, m tensile modulus and Poisson’s ra-

tio of the fiber, respectively

K, G bulk modulus and shear modulus

of the fiber, respectively

E0, m0 initial (at t = 0) tensile modulus

and Poisson’s ratio of the fiber,

respectively

R, g material viscoelastic constants for

the Maxwell model

P1, P2, Q1, Q2, P, Q linear differential operators

F axial force (threadline tension)

r, e stress and strain, respectively

ur displacement in the r-direction

REFERENCES

1. A. Pace, U.S. Patent 2,578,899 (1949).

2. A.P. Aneja, Text. Res. J., 74, 365 (2004).

3. A.P. Aneja and Y. Wang, J. Tex. Inst., 98, 127 (2007).

4. A.P. Aneja, J. Appl. Polym. Sci., 97, 123 (2005).

5. T. Kawaguchi, J. Appl. Polym. Sci., V, 482 (1961).

6. A.B. Thompson, J. Polym. Sci., 34, 741 (1959).

7. P. Chandran and S.A. Jabarin, Proc. 49th Ann. Tech. Conf.(ANTEC 91), Soc. Plast. Eng., 37, 880 (1991).

8. W.N. Findley, U.W. Cho, and J.L. Ding, J. Eng. Mat. Tech.,101, 365 (1979).

9. W.N. Findley, J.S. Lai, and K. Onaran, Creep and Relaxa-tion of Nonlinear Viscoelastic Materials, North-Holland

Publishing, Amsterdam (1976).

10. H. Yasuda and V. Stannett, J. Polym. Sci., 57, 907 (1962).

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2010 1779