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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