Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
EXPERIMENTAL AND ANALYTICAL
INVESTIGATION OF
NONISOTHERMAL VISCOELASTIC
GLASS FIBER DRAWING
by
Xiaoyong Lu
A dissertation submitted in partial ful�llmentof the requirements for the degree of
Doctor of Philosophy(Mechanical Engineering)
in The University of Michigan1999
Doctoral Committee:
Assistant Professor Ellen M. Arruda, ChairpersonAssociate Professor William W. Schultz, Co-ChairpersonAssociate Professor John W. HolmesAssociate Professor David C. MartinAssociate Professor David Mead
c Xiaoyong Lu 1999All Rights Reserved
To my wife Ying and my daughter Yiyang
ii
ACKNOWLEDGEMENTS
I wish to express my gratitude to my advisor, Professor Ellen M. Arruda, for her
supervision, inspiration and encouragement throughout the course of this project and
for her guidance and support during the period of my graduate studies. I would also
like to thank Professor William W. Schultz, for his guidance and instructions and
constant encouragement in every stage of the project, especially in the theoretical
modeling aspect. I also wish to thank Professor David Mead and Professor David
C. Martin for serving on my thesis committee and their inspiration advice and also
for allowing me to use their equipment freely. I thank Professor John W. Holmes for
serving on my thesis committee. I would also like to thank Professor R. E. Robertson
for allowing me to use his equipment for my tests.
I would like also to thank my fellow colleagues and friends, Phil and Paris for
their help and discussions in my four year study and work. I am grateful for the
�nancial support provided by the National Science Foundation.
My special gratitude goes to my wife Ying and my daughter Yiyang for their sup-
port and sacri�ce, without them, I could not have succeeded. I can never adequately
express my thanks to them for all the help and encouragement they have given to
me.
iii
TABLE OF CONTENTS
DEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii
ACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iii
LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi
LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xi
LIST OF APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii
CHAPTER
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 Review of Experimental Work . . . . . . . . . . . . 41.1.2 Review of Fiber Drawing Modeling . . . . . . . . . 6
1.2 Objectives of the Research . . . . . . . . . . . . . . . . . . . 131.3 The Organization of the Dissertation . . . . . . . . . . . . . . 14
II. EXPERIMENTAL PROCEDURES AND RESULTS . . . . . 16
2.1 Drawing Apparatus . . . . . . . . . . . . . . . . . . . . . . . 162.2 Glass Creep Measurements . . . . . . . . . . . . . . . . . . . 192.3 Fiber Property Measurements . . . . . . . . . . . . . . . . . . 21
2.3.1 Fiber Diameter . . . . . . . . . . . . . . . . . . . . 212.3.2 Birefringence . . . . . . . . . . . . . . . . . . . . . . 252.3.3 Fiber Tensile strength . . . . . . . . . . . . . . . . . 26
2.4 The E�ect of Drawing Parameters on the Maximum Draw Ratio 272.5 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 The E�ect of Heat Treatment on Fiber Tensile Strength . . . 302.7 The E�ect of Drawing Parameters on Fiber Birefringence . . 322.8 Fiber Birefringence Relaxation . . . . . . . . . . . . . . . . . 36
III. FIBER DRAWING MODELING . . . . . . . . . . . . . . . . . 40
iv
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 413.3 Fiber Drawing Modeling . . . . . . . . . . . . . . . . . . . . . 473.4 The E�ect of Viscoelasticity . . . . . . . . . . . . . . . . . . . 523.5 The E�ect of Draw Ratio . . . . . . . . . . . . . . . . . . . . 533.6 The E�ect of Draw Temperature . . . . . . . . . . . . . . . . 563.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
IV. RELAXATION MODELING . . . . . . . . . . . . . . . . . . . . 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Stretched Exponential Modeling . . . . . . . . . . . . . . . . 734.3 Je�rey Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 774.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
V. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . 83
APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86
BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 113
v
LIST OF FIGURES
Figure
1.1 A schematic diagram of ori�ce (a) and preform (b) �ber drawing. . 3
1.2 Gently convergent melt zone of glass containing stained glass owmarkers from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Sharply convergent melt zone of glass containing stained glass owmarkers from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 A typical relationship between the speci�c volume and temperatureof glass [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Schematic of �ber drawing apparatus. (All dimensions are in mm.) . 17
2.2 Two cases of temperature history of the heater . . . . . . . . . . . . 18
2.3 The heater temperature (in C) along the spinline. The temperaturemeasurements extend 10mm above and 10mm below the furnace. . . 20
2.4 Schematic of creep test �xture used to measure �ber viscosity. . . . 22
2.5 The viscosity versus temperature relation for Borosilicate glass. The�lled circles are creep test results and the solid line is from [2] . . . 23
2.6 Measured diameter variation along the length of a 120 cm �ber drawnat Tm=1215C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Glass �ber birefringence measurement set-up. . . . . . . . . . . . . 26
2.8 Glass �ber tensile strength measurement �xture. . . . . . . . . . . . 27
2.9 The e�ect of drawing temperature on the maximum draw ratio. . . 29
2.10 The heat-up time history of the �bers in the furnace for birefringencemeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
vi
2.11 The e�ect of annealing temperature on the �ber Weibull mean tensilestrength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.12 The variation of the as-drawn glass �ber birefringence with draw-ing temperature Tm and cooling rate at constant draw ratio E =ww=wi = 4410. Cooling rate is increased by increasing wi and ww
by the same amount to maintain a constant draw ratio. (i): highcooling rate wi = 0:048mm=s, (ii): low cooling rate wi = 0:027mm=s. 34
2.13 The variation of as-drawn �ber birefringence with draw ratio at con-stant draw temperature (i) Tm = 1150C, (ii) Tm = 1215C. . . . . . . 35
2.14 The birefringence relaxation of glass �ber drawn at Tm = 1215C(E = 4410) for various annealing times and temperatures. Annealingtemperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C, (v) 511C. . 37
2.15 Birefringence relaxation of glass �ber annealed at 309C from Figure2.14 including longer times. . . . . . . . . . . . . . . . . . . . . . . 38
2.16 The birefringence relaxation of glass �ber drawn at Tm = 1150C(draw ratio=4410) for various annealing times and temperatures.Annealing temperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C,(v) 511C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 The cross-section view of the preform drawing system. . . . . . . . . 43
3.2 Viscosity-temperature correlation for borosilicate glass [3] The lineis the Walther correlation: exp(exp(18:0 � 2:18 ln �)); dashed line:simple exponential: 1:3293 � 1014 exp(�0:01769�); dotted line: Ar-rhenius correlation: exp(�22:8217 + 43113=�). . . . . . . . . . . . . 45
3.3 Comparison of the dimensionless axial temperature pro�les of theglass �ber (Tg = 565C) and the environment, solid line �env, dashedline �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 The �ber radius change along the spinning line. . . . . . . . . . . . 50
3.5 The dimensionless axial and radial stress variation along the spinline. 51
3.6 The dimensionless Kelvin strain distribution. . . . . . . . . . . . . . 52
3.7 The e�ect of b on the dimensionless velocity w. . . . . . . . . . . . . 53
vii
3.8 The e�ect of b on the dimensionless axial stress Szz. . . . . . . . . . 54
3.9 The e�ect of the draw ratio E on dimensionless velocity w. . . . . . 55
3.10 The e�ect of the draw ratio E on dimensionless axial stress. . . . . 56
3.11 The e�ect of the draw ratio E on dimensionless radial stress. . . . . 57
3.12 The e�ect of the draw ratio E on dimensionless radius. . . . . . . . 58
3.13 The e�ect of the draw ratio E and draw temperature Tmax on theKelvin strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.14 The e�ect of the dimensionless drawing force on Kelvin strain atTmax = 1215C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.15 The e�ect of the draw temperature Tmax on dimensionless velocity,E = 4410. Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . 62
3.16 The e�ect of the draw temperature Tmax on axial stress, E = 4410.Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . . . . . . . 63
3.17 The e�ect of the draw temperature Tmax on radius stress, E = 4410.Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . . . . . . . 64
3.18 The e�ect of the drawing temperature Tmax on neck-down region,E = 4410. Tmax=1100, 1150, 1200 and 1215C . . . . . . . . . . . . . 65
3.19 The e�ect of the draw temperature Tmax and feed speed on Kelvinstrain, E = 4410. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.20 The e�ect of the draw temperature Tmax and feed speed on scaledKelvin strain, E = 4410. . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (13) with b = 0:5, and (i)T = 309C, � = 1000min; (ii) T = 360C, � = 150min; (iii) T = 387C,� = 50min; (iv) T = 408C, � = 20min; (v) T = 511C, � = 5min. . . 74
4.2 Fictive temperature modeling of relaxation. (a) Temperature vari-ation during annealing: (i) Fictive temperature Tf and (ii) �bertemperature T ; (b) Relaxation time evolution; and (c) Birefringencerelaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
viii
4.3 Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (4.13) with b = 1, and (i)T = 309C, � = 50min, c(T ) = 0:65; (ii) T = 360C, � = 47min,c(T ) = 0:35; (iii) T = 387C, � = 45min, c(T ) = 0:05; (iv) T = 408C,� = 16:7min, c(T ) = 0:0; (v) T = 511C, � = 8:24min, c(T ) = 0:0. . . 78
4.4 Two Je�rey elements in parallel . . . . . . . . . . . . . . . . . . . . 79
4.5 �11 versus annealing temperatures. . . . . . . . . . . . . . . . . . . 80
4.6 Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using the two Je�rey elements inparallel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.1 Variation of steady-state stress vs. strain rate for a soda-lime-silicaglass at T = 596C showing deviation from Newtonian behavior athigh strain rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.2 Schematic representation of the uniaxial stress vs. strain responseof inorganic glass near Tg at various constant applied strain rates. . 89
A.3 Schematic two-dimensional representation of the structure of a hy-pothetical compound A2O3 . . . . . . . . . . . . . . . . . . . . . . . 93
A.4 Two-dimensional representation of the structure of a modi�ed oxideglass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.5 Schematic illustration of glass melts under shear . . . . . . . . . . . 96
A.6 Stress vs. time response of soda-lime silicate glass tension tests atvarious strain rates at T=836K. . . . . . . . . . . . . . . . . . . . . 98
A.7 Stress vs. time response of soda-lime silicate glass tension tests atvarious strain rates at T=866K. . . . . . . . . . . . . . . . . . . . . 99
A.8 Stress vs. time response of soda-lime silicate glass for tension testsat various strain rates at T=836K. Curves are model simulationsusing (A.9), and symbols are the data from [4]. . . . . . . . . . . . . 100
A.9 Stress vs. time response of soda-lime silicate glass for tension testsat various strain rates at T=866K. Curves are model simulationsusing (A.9), and symbols are the data from [4]. . . . . . . . . . . . . 101
ix
A.10 Stress vs. time response of soda-lime silicate glass for compressiontests at various displacement rates at T=869K. [5] and comparisonwith the predictions using (A.9). . . . . . . . . . . . . . . . . . . . . 103
A.11 Stress vs. time response of soda-lime silicate glass for compressiontests at various displacement rates at T=902K. [5] and comparisonwith the predictions using (A.9). . . . . . . . . . . . . . . . . . . . . 104
A.12 Stress vs. time response of soda-lime silicate glass for compressiontests at various displacement rates at T=930K. [5] and comparisonwith the predictions using (A.9). . . . . . . . . . . . . . . . . . . . . 105
A.13 Stress vs. time of soda-lime silicate glass for various strain rates atT=930K. [5] and comparison with the prediction using (A.10) and(A.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.14 Stress vs. time response of soda-lime silicate glass in tension for var-ious strain rates at T=836K. Curves are Marrucci model simulationsusing (A.12)-(A.15), and symbols are the data from reference [4]. . . 110
x
LIST OF TABLES
Table
3.1 Glass and air properties . . . . . . . . . . . . . . . . . . . . . . . . 49
A.1 Parameters used in tension simulations . . . . . . . . . . . . . . . . 99
A.2 Parameters used in compression simulations . . . . . . . . . . . . . 102
xi
LIST OF APPENDICES
Appendix
A. CONSTITUTIVE THEORIES OFGLASS NEAR THE GLASS TRAN-SITION RANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A.1 A Viscoplastic Theory . . . . . . . . . . . . . . . . . . . . . . 93A.1.1 Physical Description of Oxide Glasses . . . . . . . . 93A.1.2 The Constitutive Formation . . . . . . . . . . . . . 96A.1.3 Model Prediction for Elongational Tests . . . . . . . 97A.1.4 Model Prediction for Compression Tests . . . . . . . 102
A.2 Viscoelastic Theories . . . . . . . . . . . . . . . . . . . . . . . 107A.2.1 White-Metzner Model . . . . . . . . . . . . . . . . . 107A.2.2 Marrucci Network Model . . . . . . . . . . . . . . . 109
A.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xii
CHAPTER I
INTRODUCTION
The continuous stretching of viscous molten polymer and glass to form �bers is
important in the textile and telecommunication industries. The current market for
glass �ber reinforcing materials of all types is about 1.8 million tons [6]. Glass �ber
composites are becoming environmentally friendly now that they can be recycled.
Recent advances in �ber-optic technology will continue to increase the worldwide
demand for �ber products. At the same time, the competition in the industry requires
glass �ber manufacturers to raise the e�ciency of production and quality of their
products. Better understanding and prediction of the product properties, such as
�ber tensile strength and anisotropic structure are important.
It is widely known that materials in the form of �bers have mechanical and
physical properties that are substantially di�erent from those in their bulk forms.
For instance, �laments typically have signi�cantly higher tensile strengths than in
the bulk form. The optical properties of the �ber and the bulk also di�er to some
extent. Many factors may contribute to these di�erences, but an important factor
may be the unique temperature-strain history of the �ber manufacturing process.
The �ber forming process usually involves extruding the melt through an ori�ce
and then drawing the melt in air using a high-speed winder. Preform �ber drawing
1
2
is another �ber drawing technique, often used in optical �ber manufacturing. A
schematic diagram of the two processes is shown in Figure 1.1. During melt spinning
molten polymer or glass is extruded through a bushing into cross- owing ambient
air or water spray. The solidi�ed material is taken up at a higher speed than the
velocity at the ori�ce resulting in the formation of a �lament. The preform �ber
drawing process involves feeding a solid glass rod into an axisymmetric furnace. The
softened glass rod is drawn into a �ber by a pair of winders rotating at a signi�cantly
higher speed than that of the feed rollers.
During the drawing process the material undergoes severe thermal gradients and
large mechanical deformations. Fibers made from preforms are both heated and
cooled during drawing, while ori�ce drawn �bers usually are only cooled. Although
the e�ect of processing on the physical properties of �bers di�ers in extent and
type for di�erent categories of materials, it is clear that processing parameters are
critically important. Process parameters a�ecting �ber properties include drawing
temperature, the feed and winder velocities, �ber cooling rate, and draw ratio. Post
processing, such as applying sizing and annealing, also play an important role in
determining �ber properties.
Glass �ber drawing is a well established technique. Tensile strength, diameter
uniformity and optical properties of glass �bers are the most critical. Most previ-
ous studies focus on the e�ect of processing parameters on the glass �ber mechanical
properties such as �ber tensile strength and elastic modulus. However the anisotropic
structure change during the glass �ber drawing plays an important role in the �ber
properties. Optical measurements of the anisotropy in glass �bers are seldom re-
ported in the open literature.
3
(a) (b)
Orifice
Preform
Heater
Fiber
Fiber
Figure 1.1: A schematic diagram of ori�ce (a) and preform (b) �ber drawing.
4
1.1 Literature Review
There have been signi�cant experimental and theoretical e�orts to study the
e�ects of processing parameters on �ber properties. Most published experimental
and theoretical work has been motivated by determining �nal �ber properties of
particular interest. This thesis is concerned with the preform glass drawing process.
The following is a brief review of the e�ects of �ber drawing on properties.
1.1.1 Review of Experimental Work
Early investigations of glass �ber were interested in the relationship between
diameter and strength. The experiments of Gri�th [7] show that when glass �ber
diameter decreases from 1mm to 3�m its tensile strength increases from 170 to 3,400
MPa. Gri�th suggested that the bulk glass surface contains numerous aws as small
as 5�m and that these aws become the starting points of cracks. Flaws of such size
are greatly reduced in thin �bers during the drawing process. Gri�th hypothesized
that the increase in tensile strength of �bers might be due to the orientation of the
molecules at the surface. Many investigators [8, 9, 10] supported this mechanism
for the dependence of strength on diameter. They suggested that the origin of the
aws might be in the molecular orientation in the glass during the drawing process.
Bartenev and Bovkunenko [11] proposed that structural orientation or orientation
of strong bonds increased the strength of �ne borosilicate glass �bers. Prebus et
al. [12] found evidence of ordered structure in super�ne E and C glass �bers (0.1�m
diameter) that they interpreted as indicating ordered molecular structures of up to
100 _A in length supported by electron microscope measurement.
Other investigators have rejected the assumption of structural orientation in
drawn glass �bers. Otto and Preston [13] heated drawn borosilicate glass �ber and
5
showed that thermal relative contraction in the longitudinal and radial directions
were equal. They suggested that �bers must be isotropic and the longitudinal and
transverse bonds must be of equal strength. Otto [14] also showed that the measured
strength of a glass �ber does not depend on �ber diameter under carefully controlled
experimental conditions.
Thomas [15, 16] systematically drew E glass �bers from an ori�ce and mea-
sured the tensile strength of the pristine �bers in an environment not exceeding 40%
relative humidity. He concluded that the strengths of pristine E glass �bers were
independent of the �ber diameter over the range of 5.3-15�m provided the molten
glass temperature was su�ciently high to permit a �ber of uniform diameter to be
produced. He also determined that the strength was independent of the �ber drawing
temperature (in the range of 1200 to 1340C). Thomas [16] later studied the strength
of borosilicate glass in the form of �bers and rods. Although his results showed that
the mean strength of glass rods with diameters of approximately 1.27mm was about
60% of that of �bers with a diameter about 50.8�m, he concluded that since the
maximum strength of glass rods was similar to that of glass �bers, there was little
di�erence between the intrinsic strength of glass rods and �bers. Thomas attributed
the di�erence in mean strength to the assumption that the rods contain sources of
weakness not present in the �bers. However, the draw ratio and �ber diameter in
the Thomas study are signi�cantly di�erent than those in the studies of others in
which large �ber strength changes were noted.
More recent works have shown that the physical properties of glass �bers have
a great deal to do with the processing parameters. Not only does the strength of
a glass �ber depends on its diameter, but its structure shows anisotropic properties
in the axial versus transverse directions of the �ber. Loewenstein and Dowd [17]
6
studied the tensile strengths of E glass, A glass, and a high-strength glass drawn from
an ori�ce. Their results indicate that all glasses show a tensile strength/diameter
relationship at a constant rate of loading, and drawing temperature is also likely
to be a signi�cant factor. Pahler and Br�uckner [18] determined the tensile strength
and elastic constants for pristine E glass �bers and both alkaline earth and alkali
metaphosphate glass �bers drawn from an ori�ce under well de�ned conditions. They
suggested that the Young's modulus, shear modulus and Poisson's ratio gave direct
evidence of orientation and anisotropy of the �ber structure when compared with
the isotropic structure of bulk glass.
Stockhorst and Br�uckner [19, 20] investigated the e�ect of processing parameters
on the physical properties of E glass and phosphate glass �bers drawn from an ori�ce.
By measuring the birefringence of the �bers, they concluded that glass �bers (E
glass) produced at di�erent temperatures but with the same drawing stress showed
an increasing optical anisotropy with increasing temperature. They also showed that
the alkali metaphosphate glass �bers had high birefringence values up to 104nm/mm,
indicating a high degree of orientation of linear phosphate chains.
1.1.2 Review of Fiber Drawing Modeling
Most of the previous theoretical studies of the �ber drawing process focus on
the ori�ce drawing method. In this method, glass melt is generally considered to
be Newtonian. The ow is usually taken to be one dimensional, such that axial
velocity and pressure are independent of the radial coordinate. A general evaluation
of the nonisothermal spinning process has been given by Andrews [21] and Kase
and Matsuo [22]. They have derived a set of simultaneous one-dimensional partial
di�erential equations for both steady-state and transient spinning conditions.
7
Glicksman [23] assumed the glass melt to be Newtonian with a viscosity varying
exponentially with temperature. Since most of the liquid jet attenuation occurs in
the region where the absolute value of the slope of the �ber free surface boundary
is less than one tenth, Glicksman directly formulated a one-dimensional momentum
balance equation. He concluded that the assumption of one-dimensional ow yields
excellent predictions for the behavior of a variable viscosity jet in the region where
the slope of the jet surface is less than one tenth. Matovich and Pearson [24] were
also among the �rst who derived the one-dimensional governing equations carefully.
They set up an expansion procedure, however they did not de�ne a small parameter
for the asymptotic expansion.
Schultz and Davis [25] carried out an asymptotic expansion analysis of the isother-
mal �ber drawing process for Newtonian uids. They applied lubrication type scaling
ideas to the governing equations and developed all expansions in powers of the ra-
tio of the radial to the axial-length scale. Two-dimensional numerical analysis and
visualization experiments have also shown that the one-dimension assumption is
acceptable in both ori�ce and preform �ber drawing. Huynh and Tanner [26] ana-
lyzed the nonisothermal �ber drawing process using a two-dimensional �nite-element
method. They assumed the melt to be Newtonian and found that the ow �eld is
predominantly one-dimensional except within one diameter of the ori�ce.
Geyling and Homsy [1] used visualization experiments to demonstrate that the
assumption of quasi one-dimensional extensional ow, commonly accepted for �bers
drawn from an ori�ce, also applies to �bers drawn from preforms. High temperature
ow markers consisting of stained glass were embedded in the longitudinal mid-
plane of a preform. The preform rod was assembled from two half rounds with
optically at mating surfaces, one of which had the ow markers a�xed by scoring
8
Figure 1.2: Gently convergent melt zone of glass containing stained glass ow mark-ers from [1].
and staining or by printing. The two halves were joined by fusing them at the
softening temperature. Figures 1.2 and 1.3 show the visualization results for gently
convergent and sharply convergent melt zones [1]. The ow markers indicate the
stretch history of a particular uid. Since the ow markers in the neck-down region
show relatively undistorted images, it is assumed that the ow is approximately
one-dimensional elongational ow.
Shah and Pearson [27, 28] studied the nonisothermal �ber spinning process for
Newtonian viscous uids with inertia, surface tension, and gravity e�ects included.
They also examined the e�ects of these factors on the stability of the drawing process
with a parameter representing convective heat transfer. Later Pearson and Shah [29]
extended the nonisothermal analysis to �ber spinning of power law uids. They
studied the relationship among the critical extension ratio, the power-law index, and
the shear thinning e�ect. Paek and Runk [30] modeled the nonisothermal preform
�ber drawing of a Newtonian uid. Convective and radiation heat transfer were con-
sidered in the model. The model predicted the preform neck shape and temperature
distribution within the neck-down region during the drawing of a high silica rod into
a �ber and compared with the measured neck shape.
9
Figure 1.3: Sharply convergent melt zone of glass containing stained glass ow mark-ers from [1].
10
Since the drawing process involves severe thermal gradients for either ori�ce or
preform �ber drawing, isothermal and Newtonian assumptions have little practical
signi�cance. The glass traverses from a Newtonian uid, through the transition range
in which the viscoelastic behavior governs to an elastic solid.
Figure 1.4 shows the typical relationship between speci�c volume (v) and tem-
perature (T) near glass transition [2]. In the �gure, point a is the glass in liquid form
at high temperature. On cooling, the volume gradually decreases along the line abc.
Point b corresponds to Tm, the melting point of the corresponding crystal, which
may be de�ned as the temperature at which the solid and the liquid have the same
vapor pressure or have the same Gibbs free energy. If crystallization does not occur
below Tm, the liquid mass moves into the supercooled liquid state along the line bcf,
which is an extrapolation of the line abc. No discontinuities in the v-T diagram
are observed. As the temperature and hence, energy is lowered the structure of the
liquid rearranges itself into a lower volume along the line bcf. As cooling continues,
the viscosity of the glass rapidly increases. At su�ciently low temperatures, the
state line starts a smooth departure from bcf and soon becomes a near-straight line,
ending at point g when cooled rapidly, or at h when cooled slowly. The smooth curve
between the onset of the departure from the supercooled liquid line and the glassy
state is the glass transition range. It is in this range that the viscoelastic behavior
governs the glass response. Fiber drawing involves heating above the viscoelastic
range and cooling to within the glassy state.
Both viscoelastic and nonisothermal modeling are necessary for glass �ber draw-
ing. Denn and others [31] solved the equations for steady isothermal spinning of a
viscoelastic liquid for a uid model with constant shear modulus and a single constant
relaxation time. They used a slight generalization of the classical Maxwell material
11
Figure 1.4: A typical relationship between the speci�c volume and temperature ofglass [2].
12
as the constitutive equation. Gupta et al [32] studied the polymer �ber-spinning
equations using the Oldroyd B uid constitutive equation. Sridhar and Gupta [33]
conducted isothermal �ber-spinning of an 1850 ppm solution of a polyisobutylene in
polybutene. The velocity pro�les and spinline stresses were measured. The results
are compared with the Oldroyd B constitutive model simulation. Good agreement
was obtained when the e�ects of gravity were properly taken into account. Phan-
Thien [34] also used the Oldroyd B uid to model isothermal polymer �ber spinning.
Schultz [35] presented the one-dimensional �ber-spinning analysis using a general-
ized convected Maxwell model. He found that the non-Newtonian behavior of the
uid must be severely limited for the one-dimensional equations. Nguyen et al [36]
experimentally examined the isothermal �ber-spinning of M1 uid and compared
the results with the Oldroyd B constitutive model. Recently Gupta and others [37]
examined ori�ce glass �ber drawing and its stability by using the generalized upper-
convected Maxwell model. Their results showed that cooling along the spin line
strongly stabilizes the drawing process.
Although some of the previous modeling includes non-isothermal and viscoelastic
e�ects in glass �ber drawing simulations, no attempt has been made to model the
anisotropic structural change introduced by the drawing process. In this study,
we use the Je�rey model as the constitutive equation to model the nonisothermal
viscoelastic preform glass �ber drawing and its property relaxation during post-
processing. Fiber structural anisotropy and other properties are predicted using this
model.
13
1.2 Objectives of the Research
In this study, both experimental and analytical preform �ber drawing investiga-
tions are conducted. Modeling of preform glass �ber drawing will be carried out by
using a nonisothermal viscoelastic model. Polymer or glass �bers must be produced
with speci�ed end-use properties, such as �ber dimensions, elastic modulus, tensile
strength, and refractive index. It is important to understand how processing pa-
rameters a�ect �ber mechanical and physical properties, including tensile strength,
and optical and thermal properties. Since most previous studies concentrate on the
e�ects on mechanical properties, a general understanding of these relationships has
been obtained. However little work has been done on investigating the e�ect of
drawing process on �ber birefringence (i.e. the structural anisotropy). Optical char-
acteristics of �bers such as birefringence are also important properties considered in
�ber products, especially for optical �bers. The structural orientation may a�ect the
transmission of the optical signal. The e�ect of post processing on �ber birefringence
will be conducted to study relaxation of anisotropic structure and its implications
and applications to related processes.
The proposed research has the following objectives: 1) To conduct a systematic
experimental investigation of the e�ect of processing parameters on �ber birefrin-
gence using borosilicate glass preforms. Furnace temperature, feed and drawing
speeds, and draw ratio will be varied to explore the responses of the glass �ber prop-
erties. 2) To characterize the state of as-drawn and thermally treated glass �bers
by means of standard optical and mechanical characterization techniques, i.e. the
�ber birefringence measurement and tensile strength measurement, and relate these
properties to processing parameters. The optical anisotropy of glass �bers will be
14
characterized by birefringence measurements. 3) To develop a nonisothermal, vis-
coelastic ow model of the preform �ber drawing process. This model will be based
on the 1-D ow approximation. Combined conduction, convection and radiation heat
transfer will be included. The model will be used to predict the temperature and
velocity gradients, as well as the axial and radial stresses along the spin line. The
model is also used to indirectly simulate birefringence relaxation during constant
temperature annealing.
1.3 The Organization of the Dissertation
Chapter 2 details the construction of the �ber drawing apparatus, the experimen-
tal procedures, and the results. Glass �ber birefringence measurements are described
and the e�ects of various processing parameters on �ber properties, including �ber
diameter, the maximum draw ratio and �ber as-drawn birefringence, as well as the
relaxation of birefringence under various annealing conditions are studied. The ef-
fect of annealing on �ber tensile strength is also discussed in Chapter 2. Most of the
results in Chapter 2 will soon appear in the Journal of Non-Newtonian Fluid
Mechanics [38].
The analytical study of the preform glass �ber drawing process is described in
Chapter 3. Nonisothermal viscoelastic one-dimensional theory models the drawing
process. A Je�rey model is employed as the constitutive equation and solved simul-
taneously with the conservation equations.
Glass �ber birefringence relaxation is modeled in Chapter 4. Existing theories of
structural relaxation for oxide glasses to model glass �ber birefringence relaxation
are discussed. The �ctive temperature, stretched exponential, and Je�rey models are
examined. Relaxation modeling using these theories are compared with birefringence
15
results. This chapter is also described in [38].
The dissertation concludes with a summary of new �ndings and their implications
and applications to �ber drawing related processes in textile and telecommunication
industries. Suggestions and recommendations for future work are also presented
in Chapter 5. Appendix A discusses alternative constitutive theory approaches,
a viscoplastic rate activation theory and a viscoelastic uid network theory. The
comparison of these theories with experimental results is conducted in Appendix A.
CHAPTER II
EXPERIMENTAL PROCEDURES AND
RESULTS
2.1 Drawing Apparatus
Our apparatus for continuously drawing �bers from cylindrical glass preforms is
shown in Figure 2.1. A cylindrical electric resistance heater softens and melts the
glass preforms. The heater coil made from Chromel-A wire is �tted between a Mullite
tube and a fused silica core. Insulation materials are cast around the Mullite tube.
Thermocouples embedded along the heater liner monitor the temperature during
�ber drawing. The thermocouple placed in the middle of the liner is used to control
the heater temperature. The heater power is supplied by a voltage transformer
(manually controlled) and a proportional controller (DP{26, Omega).
The glass preform is fed into the heater by a pair of feed rollers controlled by a
variable speed motor with a gear motor to reduce the rotation speed. The feed rollers
are designed to feed preforms with diameter between 1mm and 10mm. The preform
is fed into the heater and is heated (by conduction, convection, and predominantly
radiation) to soften and melt the glass. The winder is driven by a variable speed
motor with PID control (Computer Boards Inc.) to insure a constant speed under
variable load drawing conditions. The air-cooled �ber is collected after the winder or
16
17
Chromel-A coil
19.4
112
feed rollerglass preform
fused silica
insulation
winderfiber
152
112
mullite
TachometerPID ControlSystem
Thermometer
Thermometer
Thermometerwith AnalogOutput
ProportionalControlSystem
ComputerData Acquisition
System
Dimensions in mm
Figure 2.1: Schematic of �ber drawing apparatus. (All dimensions are in mm.)
by wrapping the �ber around the winder. The twin cylindrical three-inch diameter
winders are made of aluminum with a rubber coating. Insulating plates at the top
and the bottom of the heater with readily modi�ed ori�ce diameters adjust the
bulk convective ow through the furnace. The diameters of the ori�ce in the top
and bottom plates used in this study are both 10mm. The drawing distance (the
distance from the bottom of the furnace to the winder) can be varied between 10cm
and 50cm by moving the winder up or down on the frame.
One high temperature case and one low temperature case of heater temperature
history are displayed in Figure 2.2. After an initial transient, the temperature (using
a proportional controller Omega DP-26) is stable with a small variation of �2C
degrees as measured at the middle point of the heater liner.
Three millimeter diameter Borosilicate glass preforms (Corning 7740; 80.6% SiO2,
18
0 10 20 30 40 50 60 70 800
200
400
600
800
1000
1200
Hea
ter
Tem
pera
ture
s (C
)
Time (min.)
Figure 2.2: Two cases of temperature history of the heater
13.0% B2O3, 4.0% Na2O, 2.3% Al2O3, 0.1% miscellaneous by weight; ASTM anneal-
ing point (glass transition temperature) 565C) were drawn at various feed and winder
speeds and a range of drawing temperatures. Borosilicate glass was used instead of
E-glass because it is a silicate glass easily obtained in rod form. An optical micro-
scope is used to measure �ber diameters and their variation under stable drawing
conditions and to compare with theoretical values determined from the draw ratio.
The draw ratio is de�ned as the ratio of drawing speed to feed speed. The e�ect of
drawing temperature on the maximum draw ratio (draw ratio at which �ber breaks)
has also been examined. A typical temperature pro�le measured by a thermocouple
traversed down the centerline of the cylindrical heater when no �ber is being drawn
19
is illustrated in Figure 2.3. The drawing temperature Tm is de�ned as the maximum
temperature along the spinline. In subsequent tests the heater was controlled by Tm
as estimated by a thermocouple placed just inside the fused silica at the approximate
height of the maximum shown in Figure 2.3. Tm was varied in the range of 1100C to
1215C to study the e�ect of drawing temperature on birefringence. The location of
the temperature maximum can be controlled by convection suppression and radiation
shields at the top and bottom or variable winding spacing. For the uniform coil
electric heater, the maximum occurs slightly higher than the heater midpoint due to
natural convection.
The preforms were usually drawn at a feed speed of wi = 0:048 mm/s, and draw-
ing speed of ww = 212:0 mm/s to produce 45�m diameter �bers (Subscripts i and w
refer to quantities at the inlet and winder, respectively.). A similar series of tests at
wi = 0:027 mm/s and ww = 119 mm/s examined the e�ect of cooling rate on birefrin-
gence. A series of tests at Tm=1150C and 1215C for wi = 0:048 mm/s was conducted
at various draw speeds to study the e�ect of draw ratio on birefringence. Fibers were
carefully collected (without sizing) after the winder and stored in dessicators prior
to preparation for annealing and/or birefringence measurements.
2.2 Glass Creep Measurements
For modeling purposes, the rheological properties (primarily viscosity) of the
borosilicate glass �bers have been measured. Low temperature (511C and 618C)
viscosities were measured by using �ber creep tests.
Figure 2.4 is a schematic of the creep test experiment. Glass beaded ends were
formed at the two ends of a thin �ber (120�m in diameter) The �ber was suspended
from a steel frame by one bead and a weight was hung from the other bead. The
20 0
200
400
600
800
1000
1200
0 20 40 60 80 100 120 140 160 180
H
ea
ter
Te
mp
era
ture
(C
)
z (mm)
Figure2.3:
Theheatertemperature
(inC)alongthespinline.
Thetemperature
measurements
extend10mm
aboveand10mm
below
thefurnace.
21
apparatus was put into a constant temperature furnace at 511C and �ber length was
measured as a function of time for a 24 hour period. This procedure was repeated
at 618C. Calipers with an accuracy of 0.01mm measured the �ber length. Using this
simple elongational (Trouton) viscometer, the viscosity of the glass was calculated
(assuming small elongation) from
� =LF
3�r2v; (2.1)
where � is viscosity, L is the initial �ber length, F = mg is the force acting on the
�ber (the mass of the weight m is 17g), r is the original �ber diameter, and v is the
average creep velocity computed from �ber elongation and time. Typical changes
in L were about 10mm at 618C and 1mm at 511C. The measured viscosities of
Borosilicate glass were 5.3�1011P at 618C and 9.3�1013P at 511C. These are within
the 5% of the published values shown in Figure 2.5 obtained using a similar method
in the low temperature range.
2.3 Fiber Property Measurements
2.3.1 Fiber Diameter
Glass �bers, especially optical glass �bers, are best produced with uniform di-
ameter under stable conditions. To examine the uniformity of �bers drawn with
the preform �ber drawing apparatus, �bers were drawn from a 3mm preform. Fiber
diameter was measured using a microscope equipped with NIH Image software. The
resolution of this measuring method is 0.32�m, and its accuracy is �1.0�m. The
typical diameter variation along the �ber length for 59�m diameter �bers is shown
in Figure 2.6. Under stable drawing conditions the preform �ber drawing apparatus
draws uniform �bers to within �1�m.
22
Glass Fiber
17g Weight
Frame
Furnace Chamber
Steel
L=40mm
120µm
Figure 2.4: Schematic of creep test �xture used to measure �ber viscosity.
23
500 600 700 800 900 1000 1100 1200 1300 140010
2
104
106
108
1010
1012
1014
1016
Temperature (C)
Vis
cosi
ty (
P)
400
Figure 2.5: The viscosity versus temperature relation for Borosilicate glass. The�lled circles are creep test results and the solid line is from [2]
24
0 20 40 60 80 100 12050
55
60
65
Distance (cm)
Dia
met
er (
µm)
Figure 2.6: Measured diameter variation along the length of a 120 cm �ber drawnat Tm=1215C.
25
2.3.2 Birefringence
It is di�cult to measure the birefringence of a single glass �ber since the re-
tardation per unit thickness of glass is quite small (approximately 10�7 to 10�5).
Br�uckner [19] used an isotropic circular quartz tube �lled with many �bers to increase
the retardation signal. The �bers in the tube were immersed in a refractive index
matching liquid. This approach was followed here but with an optically-isotropic
square quartz tube to eliminate scattering and refraction from a round tube. Fiber
bundles were collected, packed and pulled through 2mm inner thickness square quartz
tubes to increase the retardation signal. Up to �ve �ber tubes were stacked to fur-
ther increase the retardation for measurement of low birefringence. The tube and
�bers were then immersed in an index of refraction matching uid to reduce light
scattering from the �ber surfaces.
A polarizing microscope equipped with a tint plate (530nm) �rst veri�ed the
existence of optical anisotropy in glass �bers from the color variation. Quantitative
�ber birefringence was measured with a polarizing microscope out�tted with a light
intensity meter as illustrated in Figure 2.7. The axial direction of the �ber sample
is aligned at 45 degrees to the polarizer axis. The light intensity transmitted is
measured and recorded by the intensity meter. The birefringence of the sample is
calculated from
4n = ��
h�sin�1
sI
Io(2.2)
where 4n is sample birefringence, � is a calibration coe�cient of the polarizing
microscope, � is the wavelength of the monochromatic light (589.6 nm), and h is the
e�ective thickness of the glass �bers in the tube. The e�ective thickness of the glass
�bers is computed from the weight, length and density of the �bers as h =�w�l
�1=2.
26
Photometer
Light SourceFilter
PolarizerSample
Analyzer
45
Figure 2.7: Glass �ber birefringence measurement set-up.
Io is the light intensity when the polarizer, analyzer and the �ber samples are all
aligned in the same direction. I is the light intensity when the sample is rotated 45
degrees and the polarizer and the analyzer are crossed as shown in Figure 2.7.
2.3.3 Fiber Tensile strength
Tensile strength is one of the most important characteristics of glass �ber products
used as reinforcement in composites and as optical waveguides in telecommunication.
Fiber tensile strength was measured using a setup shown in Figure 2.8. A �ber
sample was mounted on a pair of pulleys with a 35mm gage separation and �xed with
adhesive tape away from the point of initial roller contact. The test was conducted
at constant displacement rate on an Instron machine (Model TMS).
27
Gage Length35 mm
PP
Fiber
Figure 2.8: Glass �ber tensile strength measurement �xture.
2.4 The E�ect of Drawing Parameters on the Maximum
Draw Ratio
Draw ratio is de�ned as the ratio of the winder drawing speed to the preform feed
speed. One of the important issues in �ber drawing is to continuously draw �bers
without draw resonance or breakage. Draw resonance is a temporal instability in �ber
spinning caused by a constant force condition [25]. A regular and sustained periodic
variation in the drawn �lament diameter occurs when draw resonance is encountered.
This phenomenon can interrupt the production process and greatly reduce outputs of
production. Variation in diameter and breakage may result from many other factors,
such as unsteady feed and winding rates, temperature perturbations in the heater,
and heating and cooling variations.
Since the ow �eld of the drawing process can be modeled as a relatively simple
quasi-one-dimensional approximation, numerous investigations have concentrated on
the analytical and numerical analysis for draw resonance. A critical draw ratio
(the draw ratio at which draw resonance occurs) has been obtained under speci�ed
drawing conditions [39]. Experimental studies, however, are relatively rare in the
28
literature.
We brie y studied the drawing instability experimentally to explore if draw res-
onance occurs at �ber breaks. Maximum draw ratios are determined for a given
heater temperature by gradually increasing winder speed and hence, the draw ratio
until the �ber breaks. The draw ratio is increased incrementally, and the drawing of
�bers is restabilized; if there is no breakage, the draw ratio is again increased and
the process is repeated until the �ber breaks. The e�ect of drawing temperature on
the maximum draw ratio is shown in Figure 2.9. Each point represents the average
of three tests.
The maximum (or critical) draw ratio we obtained is generally higher than the
critical draw ratio (draw resonance) numerically acquired for nonisothermal viscoelas-
tic ow for ori�ce ow [37]. Since draw resonance causes large diameter variations,
the diameter of the broken �ber was examined to monitor for draw resonance. The
diameter variation remained small (as in Figure 2.6) even for �bers drawn to the
highest draw ratio in this study, suggesting that no draw resonance occurs for the
preform �ber drawing. The cause of �ber failure is fracture as the tensile strength is
exceeded.
The maximum draw ratio increases with drawing temperature as shown in Figure
2.9. As drawing temperature increases glass viscosity decreases, thus the drawing
force decreases, allowing greater extension ratios before breakage. As T increases
still further, the viscosity becomes very low and a capillary instability may occur.
Further work is required in this area to determine the ultimate tensile strength as a
function of temperature and to determine why draw resonance is suppressed.
29
1020 1040 1060 1080 1100 1120 1140 11602000
2500
3000
3500
4000
4500
5000
5500
Drawing Temperature (C)
Max
imum
Dra
w R
atio
Figure 2.9: The e�ect of drawing temperature on the maximum draw ratio.
30
2.5 Post-Processing
The e�ect of post-processing, i.e. annealing on glass �ber tensile strength and
birefringence is examined experimentally. The birefringence relaxation of �bers was
studied by annealing drawn �bers for various times and temperatures. The sam-
ples were prepared as described above for birefringence measurement and annealed
at constant temperature in a furnace maintained to within �2C. Fibers were air
quenched after annealing and held in a dessicator until tested. Two sets of �bers
drawn at temperatures Tm=1150C and 1215C at a draw ratio of 4410 were studied
for the birefringence relaxation. The annealing temperatures ranged up to 511C.
The annealing time was recorded as the residence time in the furnace without regard
to heat-up or cool-down times. The heat-up and cool-down times have been mea-
sured for some annealing processes by placing a thermocouple in the middle of the
�ber bundle. These results show that after placing the specimen in the furnace, the
temperature approaches the speci�ed temperature exponentially with time constants
in the range of 10 to 30 seconds shown in Figure 2.10. Cooling after removing the
�bers from the furnace shows a similar time constant.
2.6 The E�ect of Heat Treatment on Fiber Tensile Strength
The e�ect of post-processing by annealing on the strength of glass �bers was
brie y investigated in terms of annealing temperature and time. The as-drawn glass
�bers were heat treated in a furnace for one hour. The nominal annealing tempera-
tures were 100, 300 and 500C. The Weibull [40] mean tensile strength of glass �bers
was tested using the apparatus sketched in Figure 2.8. The Weibull mean tensile
strength is expressed as Ps = exp(�( ��0)m), where Ps is the survival probability, �
is the applied stress, �0 is the stress corresponding to a survival probability of 1eor
31
0 50 100 150 200 250 300 3500
100
200
300
400
500
600
Time (s)
Fib
er T
empe
ratu
re in
the
Fur
nace
(C)
Figure 2.10: The heat-up time history of the �bers in the furnace for birefringencemeasurements.
32
37%, and m is the Weibull modulus. A �ber sample was mounted on a pair of pul-
leys with a 35mm gage separation and �xed with adhesive tape away from the point
of initial roller contact. The test was conducted at constant displacement rate on
an Instron machine (Model TMS). Fiber samples which broke at the roller contact
point were discarded. Figure 2.11 shows the results of the Weibull mean strength
and modulus m versus the annealing temperature. Each point represents 12 �ber
samples. The tensile strength is shown to decrease with an increase in the annealing
temperature. A possible reason for this decrease may be that annealing causes aws
on the surface of glass �bers, resulting in the decrease of the tensile strength. The
as-drawn strength is similar to that previously reported [41].
2.7 The E�ect of Drawing Parameters on Fiber Birefrin-
gence
Strength measurements are di�cult to perform and the data show large scatter
as expected from tensile strengths of brittle materials. Therefore we sought and
found a related measurement, birefringence, that is more readily measured and re-
peatable. Figure 2.12 shows the variation of the as-drawn �ber birefringence with
drawing temperature for two cooling rates. The birefringence of the as-drawn glass
�bers decreases with increasing drawing temperature at constant draw ratio. Each
point represents the average of three measurements with a deviation �0:03 � 10�5.
The decrease in birefringence is moderate for drawing temperatures up to 1200C;
above 1200C (near the working point, viscosity of 103Pa�s) the birefringence of the
�bers drops signi�cantly with draw temperature. This suggests that as the drawing
temperature approaches the working point the drawing tension and stress decrease
quickly, causing the rapid reduction of the birefringence. The cooling rate during
33
Weibull mean Strength (MPa)
100 200 300 400 500
Annealing Temperature (C)
200.0
300.0
400.0
100.0
m=3.2m=4.2
m=2.4
m=5.8
m: Weibull Modulus
(as drawn)
Figure 2.11: The e�ect of annealing temperature on the �ber Weibull mean tensilestrength.
34
0
2e-06
4e-06
6e-06
8e-06
1e-05
1.2e-05
1.4e-05
1100 1120 1140 1160 1180 1200 1220
Bire
frin
genc
e
Drawing Temperature (C)
(i)
(ii)
Figure 2.12: The variation of the as-drawn glass �ber birefringence with drawingtemperature Tm and cooling rate at constant draw ratio E = ww=wi =4410. Cooling rate is increased by increasing wi and ww by the sameamount to maintain a constant draw ratio. (i): high cooling rate wi =0:048mm=s, (ii): low cooling rate wi = 0:027mm=s.
drawing was varied by proportionally changing the feed speed and winder speed to
maintain a constant draw ratio. Figure 2.12 results show that at constant draw tem-
perature Tm and draw ratio E, the as-drawn birefringence increases with increasing
cooling rate. This implies that a faster cooling rate is more e�ective at \freezing"
the anisotropic elastic strains developed during drawing deformation.
The e�ect of draw ratio on �ber birefringence was examined by varying the winder
speed at constant feed speed shown in Figure 2.13. The draw temperature was
constant at Tm = 1150C and Tm = 1215C and the feed speed at wi = 0:048mm/s.
35
0
2e-06
4e-06
6e-06
8e-06
1e-05
1.2e-05
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Bire
frin
genc
e
Draw Ratio
(i)
(ii)
Figure 2.13: The variation of as-drawn �ber birefringence with draw ratio at constantdraw temperature (i) Tm = 1150C, (ii) Tm = 1215C.
Drawing speeds ww varied from 0.028m/s to 0.212m/s resulting in draw ratios E of
587 to 4410, and corresponding �ber diameters of 120�m to 45�m.
Fiber birefringence increases almost linearly with draw ratio in the range exam-
ined. At constant feed speed, the �ber diameter is inversely proportional to the
square of the drawing speed. Birefringence of glass �bers increases with increasing
draw ratio or with decreasing �ber diameter. These data con�rm the expected trend
of increasing birefringence with increasing tensile stresses or drawing forces. This
would indicate that high draw ratios and low draw temperatures are preferential in
developing birefringence.
36
2.8 Fiber Birefringence Relaxation
Figure 2.14 shows the scaled birefringence relaxation of �bers drawn at Tm =
1215C and draw ratio of 4410 (leading to an as-drawn birefringence of 4:72� 10�6)
annealed at various times and temperatures. Scaled birefringence is de�ned as the
relaxed birefringence divided by the as-drawn birefringence. Figure 2.14 includes
results for all temperatures and annealing times up to 180 minutes. Fibers were
annealed at 309C for longer times and the results are reported in Figure 2.15. At
annealing temperatures of 408C and above birefringence relaxation is rapid and com-
plete within 60 minutes annealing time. Fibers annealed at temperatures of 360C
and below appear to not fully relax as their birefringence asymptotes after 180 min-
utes. Fibers annealed at 309C for longer times show no further relaxation after 24
hours. Fibers annealed at 387C show an intermediate behavior whereby relaxation
is complete after an excess of 200 minutes.
Figure 2.16 contains the relaxation results for �bers drawn at Tm = 1150C and
a draw ratio of 4410 (leading to an as-drawn birefringence of 1:21 � 10�5). The
relaxation behavior is similar to that shown in Figure 2.14. Complete birefringence
relaxation occurs quickly at annealing temperatures of 408C and above; the birefrin-
gence does not fully relax at annealing temperatures below 360C, and intermediate
relaxation behavior is observed at 387C.
37
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
Sca
led
Bire
frin
genc
e
Time (min)
(i)
(ii)
(iii)
(iv)(v)
Figure 2.14: The birefringence relaxation of glass �ber drawn at Tm = 1215C(E = 4410) for various annealing times and temperatures. Annealingtemperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C, (v) 511C.
38
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200 1400
Sca
led
Bire
frin
genc
e
Time (min)
Figure 2.15: Birefringence relaxation of glass �ber annealed at 309C from Figure2.14 including longer times.
39
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
Sca
led
Bire
frin
genc
e
Time (min)
(i)
(ii)
(iii)
(iv)(v)
Figure 2.16: The birefringence relaxation of glass �ber drawn at Tm = 1150C (drawratio=4410) for various annealing times and temperatures. Annealingtemperatures: (i) 309C, (ii) 360C, (iii) 387C, (iv) 408C, (v) 511C.
CHAPTER III
FIBER DRAWING MODELING
3.1 Introduction
Kase and Matsuo [22] were among the �rst to analyze nonisothermal �ber spin-
ning. However, their studies did not take into account the viscoelastic behavior in
elongational ow. Glicksman [23] also assumed the glass melt to be Newtonian with
a viscosity varying exponentially with temperature. Matsui and Bogue [42] analyzed
melt spinning with a generalized nonisothermal viscoelastic theory. Recently Gupta
and others [37] conducted ori�ce glass �ber drawing and its stability by using the
generalized upper-convected Maxwell model.
It is important to choose the constitutive equation to appropriately represent the
material behavior during �ber drawing. We seek the simplest model possible that
allows for residual strain or stress to remain in drawn �ber. For completeness we ex-
amine in Appendix A several additional constitutive models that exhibit viscoplastic
and strain softening behavior found in other simple, elongational experiments [4].
In this chapter, the nonisothermal viscoelastic glass preform �ber drawing pro-
cess is analyzed. A temperature dependent viscosity is assumed, but density, shear
modulus and all other quantities are considered constant.
A generalized Je�rey viscoelastic model is used to simulate the preform glass
40
41
drawing process. The e�ect of drawing process parameters on the �ber properties are
studied. Anisotropic structural change, i.e. glass �ber birefringence, is also analyzed
using the elastic \frozen" strain due to rapid cooling. The momentum and energy
equations are solved simultaneously and the measured heater temperature pro�le is
the environmental temperature in the energy equation. The e�ect of viscoelasticity
on the �ber drawing is studied by comparing the results of Newtonian, Maxwell and
Je�rey models.
3.2 Governing Equations
Consider a cylindrical glass preform of initial radius R0 inside a heater, as shown
in Figure 3.1. The temperature pro�le of the heater is shown previously in Figure
2.3. The glass preform temperature at z = 0 is well below its transition temperature.
The point z = 0 is 10mm above the heater top ori�ce. (Since simulation shows that
most deformation occurs at around z = zm where T (zm) = Tm, it is not so important
where z = 0 is de�ned.). As the preform descends into the heater its temperature
increases until it reaches a maximum temperature Tmax close to where the maximum
heater temperature Tm occurs. The glass preform diameter begins to contract under
tension at temperatures above its softening point.
The radius of the elongating preform as a function of axial position is denoted
by R(z). As R(z) decreases, other variables such as the velocity, axial stress and
radial stress change along the spinline. The �ber starts to cool as it descends to
the lower edge of the heater. The glass �ber \freezes" into a solid at a temperature
below its transition range at z = L, a point 10mm below the heater bottom. As
with the origin, the de�nition of L is not so important as long as the temperature is
the temperature is close to the ambient temperature there. At this point, the �ber
42
is drawn past the winder at a given average velocity ww (the subscript w refers to
the quantities at the winder).
A di�erential Je�rey constitutive equation is chosen to simulate the glass �ber
drawing process in this chapter and the post processing birefringence relaxation of
glass �bers in the next chapter. The Cauchy stress tensor, T is
T = �pI+ S; (3.1)
where I is the identity tensor, p is the pressure and S is the extra Cauchy stress
tensor, given by
S + �1
DSDt � a(S �D+D � S)
!= 2�(�)
D+ �2
DDDt
!; (3.2)
Here � is the temperature dependent Newtonian viscosity, a is a rate parameter, �1
and �2 are the relaxation and retardation times, respectively, D is the deformation
rate tensor given by
D =1
2(rv +rvT ); (3.3)
where v is the velocity vector, and rv is the gradient of v; vT is the transpose of v.
and D
Dtis the Jaumann derivative such that
DSDt =
DS
Dt+
1
2(! � S� S � !); (3.4)
where ! is the vorticity tensor de�ned as
! =1
2(rv �rvT ): (3.5)
In the following model, the shear modulus of the glass is assumed constant unless
otherwise noted. Hence, the relaxation time �1 = �=G has the same temperature
dependence as the viscosity in accordance with the thermo-rheological simplicity
43
0
z=0
z=L
R(z
)R
r
z
0
200
400
600
800
1000
1200
0 20 40 60 80 100 120 140 160 180
H
eate
r T
empe
ratu
re (
C)
z (mm)
Figure
3.1:
Thecross-sectionviewof
thepreform
drawingsystem
.
44
(TRS) assumption [43]. For simplicity the retardation time �2 is also assumed to be
proportional to �1, �2 = b�1, where b is a constant.
The exponential temperature-viscosity correlation is widely used in glass �ber
drawing modeling [23]; the exponential correlation typically �ts viscosity data in the
moderate to high temperature range. An exponential correlation does not capture
the temperature dependence of borosilicate glass over a broad temperature range.
Although an Arrhenius function is valid for all temperatures (other than absolute
zero), it did not �t the data well in Figure 3.2. The often used modi�ed Arrhenius
model, where � is replaced with �� �ref �ts higher temperature data but is singular
at � = �ref and �ref is usually larger than ambient temperature. A Walther tempera-
ture viscosity correlation involves only two parameters [44] and �ts the experimental
viscosity data for borosilicate glass over a wide range of temperatures. The Walther
viscosity-temperature correlation is given by
� = �0 exp(exp(v0 � v1 ln �)); (3.6)
where � is the Newtonian viscosity and �0, v0 and v1 are correlation parameters, and
� is the temperature in degrees Kelvin. (Since dimensionless governing equations are
solved, �0 is canceled out when scaling the equations.) This correlation is better for
our purpose because it is not singular except at absolute zero temperature. Equation
(3.6) with v0 = 18:0 and v1 = 2:18 is compared with the data [3] in Figure 3.2. and
found to �t the data well over a large temperature range. The Walther correlation
is used to describe the temperature dependent viscosity in the following modeling.
The heat transfer coe�cient including convection and radiation is chosen in
lumped form [22] as
h =k12R
C
2�1wR
�1
!m
; (3.7)
45
600 800 1000 1200 1400 1600
105
1010
1015
1020
1025
1030
1035
1040
Temperature (K)
Vis
cosi
ty (
P)
Figure 3.2: Viscosity-temperature correlation for borosilicate glass [3] The line is theWalther correlation: exp(exp(18:0�2:18 ln �)); dashed line: simple expo-nential: 1:3293�1014 exp(�0:01769�); dotted line: Arrhenius correlation:exp(�22:8217 + 43113=�).
46
where k1, �1, and �1 are the thermal conductivity, density, and the viscosity of
the ambient air at room temperature, respectively. The coe�cients C and m are
determined from experiments [45]. To combine the e�ects of convection and radiation
in the heat transfer coe�cient, the constants C and m are modi�ed from those in
Kase and Matsuo [22].
Lubrication scaling [25] of the governing equations is employed for all expansions
in powers of � = Ri=L. The following dimensionless variables are de�ned:
R�=
RRi, z� = z
L , u�=
u�wi
, w�=
wwi, �� = �
�m
�� = ���1�m��1
, G�=
Gwi�m=L
, T� = T
wi�m=L
where the superscript * refers to the nondimensional quantities, and it is dropped
in the future equations for notational convenience. The subscript i represents the
initial value at z = 0, G is the shear modulus of the borosilicate glass, and �m is the
viscosity of the glass corresponding to the maximum temperature Tmax. The Peclet
and Biot numbers are de�ned as follows:
Pe =�wiRiCp
k�
Bi = 1
�2hRi
k
where �, Cp, and k are the density, speci�c heat, and thermal conductivity of borosil-
icate glass, respectively. The Peclet number Pe is an inverse dimensionless thermal
conductivity and the Biot number is a dimensionless heat transfer coe�cient. These
parameters are assumed to be constant for simplicity. Glass properties in the transi-
tion range are chosen for the model simulation. By requiring the Biot number to be
scaled in the above manner with Bi = O(1), the leading-order solution for � depends
47
only on the axial coordinate consistent with the one-dimensional assumption.
The unsteady one-dimensional dimensionless mass and momentum conservation
equations are obtained from the leading-order equations following the procedure of
Gupta et al [37].
@
@t(R2) +
@
@z(wR2) = 0; (3.8)
@
@z[R2(Tzz � Trr)] = 0 (3.9)
The unsteady energy equation includes axial conduction, advection and heat
transfer through the �ber surface to the environment due to convection. The dimen-
sionless energy equation gives:
Pe
@�
@t+ w
@�
@z
!=
1
R2
@
@z
R2@�
@z
!� 2Bi
R(� � �env(z)); (3.10)
where �env(z) is the dimensionless temperature of the environment. The above di-
mensionless conservation laws for mass, momentum, and energy along with the con-
stitutive equation relation (3.2) form the governing set of equations for the glass �ber
drawing analysis.
3.3 Fiber Drawing Modeling
This study only considers the steady-state drawing process; the governing equa-
tions are simpli�ed by removing the partial derivative with respect to time. The
mass conservation equation becomes:
R =1pw
(3.11)
The momentum conservation equation is simpli�ed to
Szz � Srrw
= F; (3.12)
48
where F is the dimensionless axial force independent of z, and the axial energy
conservation equation is
d2�
d2z� 1
w
dw
dz+ Pe w
!d�
dz� 2pwBi(� � �env) = 0; (3.13)
The constitutive equation (3.2) for elongational ow is simpli�ed to
�1w@Szz@z
+
1� 2a�1
@w
@z
!Szz � 2�
@w
@z� 2��2w
@2w
@z2= 0; (3.14)
�1w@Srr@z
+
1 + a�1
@w
@z
!Srr + �
@w
@z+ ��2w
@2w
@z2= 0; (3.15)
for the axial and radial directions, respectively.
There are �ve unknowns �(z), w(z), Szz(z), Srr(z) and R(z) in the above system
of equations. The boundary conditions are given as follows:
w = 1; R = 1; � = �env at z = 0;
w = E;d2�
dz2= 0 at z = 1; (3.16)
where E is the draw ratio. An upstream stress boundary condition completes the
problem statement. At z = 0, the momentum equation becomes Szz � Srr = F and
a combination of Szz =23F and Srr = �1
3F is employed as the boundary conditions
as in Schultz [25].
For the example presented below, the diameter of the glass preform is Ri =
3�10�3m. The feed speed is wi = 4:8�10�5m/s, the maximum drawing temperatures
is Tmax = 1215C, and T1 is 20C. The glass and air properties are listed in Table
3.1. The draw ratio was nominally taken as E = 4410. The values of C and m in
(3.7) are chosen to be 1.117 and 0.13, respectively [37].
Figure 3.3 shows the predicted dimensionless temperature pro�le of the glass �ber
and heater along the spinline. The environmental (heater) temperature pro�le is
49
Table 3.1: Glass and air properties
�(g=m3) cp(J=kg �K) k(w=m �K) G(GPa) �1(Pa � s)
Glass 2:23� 103 1:45� 103 1.31 26.67
Air(20C) 0.0392 1:11� 103 0.0616 36:85� 10�6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Z
Dim
ensi
onle
ss T
empe
ratu
re
Figure 3.3: Comparison of the dimensionless axial temperature pro�les of the glass�ber (Tg = 565C) and the environment, solid line �env, dashed line �.
50
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.2
0.4
0.6
0.8 1
1.2
Z
Radius R
Figu
re3.4:
The�ber
radiuschange
alongthespinninglin
e.
measu
redas
describ
edin
Chapter
2.The�ber
temperatu
repro�
leisapprox
imately
thesam
eas
�en
v .This
suggests
that
convection
andrad
iationheat
transfer
are
su�cien
tto
instan
taneou
slyheat
the�ber.
Thus,to
furth
ersim
plify
theanaly
sis,�
isset
equal
tothegiven
�en
vandtheenergy
equation
canbediscard
ed.
Figu
res3.4
disp
laysthe�ber
radiusvariation
intheneck
-dow
nregion
.The
glasspreform
radiuschanges
littleuntil
itreach
esthesoften
ingpoin
ttem
peratu
re.
Most
ofthedeform
ationoccu
rsnear
Tmax .
Asthetem
peratu
redecreases,
the�ber
radiusgrad
ually
decreases
until
thelarge
viscosity
makes
deform
ationnolon
ger
possib
le.Figu
res3.5
show
sthedim
ension
lessaxial
andrad
ialstress
distrib
ution
alongthespinlin
e.Therad
ialstress
isalm
ostnegative
one-h
alfof
theaxial
stress
asin
New
tonian
elongation
al ow
.Both
stressesapproach
constan
tas
theheater
51
00.1
0.20.3
0.40.5
0.60.7
0.80.9
110
1
102
103
104
105
106
Z
Axial Stress and Radial Stress
Szz
- Srr
Szz S rr
Figu
re3.5:
Thedim
ension
lessaxial
andrad
ialstress
variationalon
gthespinlin
e.
temperatu
redrop
sbelow
thesoften
ingpoin
t(at
z=0:7)
astheglass
�ber
e�ectively
becom
esan
elasticsolid
.
Tomodeltheanisotrop
icglass
structu
ralchange
durin
g�berdraw
ing,thedim
en-
sionless
strainin
theKelv
inelem
ent(th
e`Kelv
instrain
')issim
ulated
bysep
arating
itfrom
thetotal
strain.Thedi�eren
tialequation
fortheKelv
instrain
isgiven
as
@�1
@z=
Szz �
Srr
��
�1
�2 ;
(3.17)
where
�1isthedim
ension
lessstrain
intheKelv
inelem
ent.
This
equation
canbe
solvedas
apost-p
rocessin
gstep
toobtain
theKelv
instrain
,indicative
ofanisotrop
ic
structu
reandhence
birefrin
gence.
Figu
re3.6
show
stheKelv
instrain
alongthe
spinlin
e.It
isfrozen
into
the�ber
asthetem
peratu
redrop
sto
room
temperatu
re.
ThisKelv
instrain
may
relaxas
temperatu
reincreases
durin
gpost-p
rocessin
g,while
52
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.5 1
1.5 2
2.5 3x 10
−5
Z
Kelvin Strain ε1
Figu
re3.6:
Thedim
ension
lessKelv
instrain
distrib
ution
.
thelarge
deform
ationinduced
bytheviscou
s ow
does
not
recover.
3.4
TheE�ectofViscoelastic
ity
Thee�ect
ofvisco
elasticityon
thedraw
ingprocess
isstu
died
bycom
parin
gthe
New
tonian
,Maxwell,
andJe�
reymodels.
TheJe�
reyequation
becom
esMaxwell
equation
as�2 !
0andto
New
tonian
ow
when
both
therelax
ationandretard
ation
times
approach
zero.Theratio
ofretard
ationtim
eto
relaxation
timeb=�2 =�
1is
chosen
tobevery
small
tosim
ulate
Maxwell
ow
.Theran
geof
bbetw
een10�15
and10�1isexam
ined
toanaly
zetheKelv
in-elem
ente�ect.
For
thegiven
draw
ing
param
eters,theoverall
draw
ingvariab
lesof
velocity,
axial
andrad
ialstress,
and
�ber
radiusshow
littledi�eren
ceas
inFigu
res3.7
and3.8.
Thissuggests
that
since
53
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
500
1000
1500
2000
2500
3000
3500
4000
4500
Z
Velocity w
b=10
-15
b=10
-1
Figu
re3.7:
Thee�ect
ofbon
thedim
ension
lessvelo
cityw.
most
ofthedeform
ationoccu
rsinthehigh
temperatu
reregion
,thematerialb
ehavior
durin
gdraw
ingise�ectively
New
tonian
.Thevisco
elasticityhas
littlee�ect
onthe
overallresp
onse
before
the�ber
iscooled
toasolid
.How
ever,theJe�
reymodelis
required
tosim
ulate
theperm
anentanisotrop
icstru
ctural
change
ofthesolid
�ber.
This
realizationis
importan
tsin
cemost
visco
elasticmodels
of�ber
spinningare
variationsof
Maxwell
models.
3.5
TheE�ectofDrawRatio
Thee�ect
ofthedraw
ratioon
draw
ingvariab
lesisanaly
zedin
thissection
.The
draw
ingvariab
lesinclu
de�ber
velocity,
axial
andrad
ialstresses,
�ber
radiusandthe
frozen-in
strainintheKelv
inelem
ent.Thefrozen
-instrain
represen
tstheanisotrop
ic
54
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.5 1
1.5 2
2.5 3
3.5 4x 10
5
Z
Axial Stress Szz
b=10
-15
b=10
-1
Figu
re3.8:
Thee�ect
ofbon
thedim
ension
lessaxial
stressSzz .
55
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
500
1000
1500
2000
2500
3000
3500
4000
4500
Z
Vel
ocity
E=587
E=1150
E=3200
E=4410w
Figure 3.9: The e�ect of the draw ratio E on dimensionless velocity w.
structural change during the drawing process, characterized by birefringence. The
draw ratios are chosen as E=587, 1150, 3200 and 4410, corresponding to those of our
experiments. The dimensionless velocity pro�les are shown in Figure 3.9. For a given
draw ratio, the dimensionless velocity starts as w = 1 at z = 0 and increases rapidly
in the high temperature region near z = 0:45 where T = Tmax. As the temperature
decreases the velocity gradient tends to zero.
Figures 3.10 and 3.11 display the dependence of the axial and radial stresses
on the given draw ratios. Both the absolute axial and radial stresses increase with
an increase in the draw ratio, while they have the opposite signs, i.e., positive ax-
ial stress and negative radial stress. These stresses increase as E increases with a
corresponding decrease of �ber diameters shown in Figure 3.12. The dependence
56
10
10
10
10
10
10
1
2
3
4
5
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Z
Axi
al S
tres
s
E=4410
E=3200
E=1150
E=587
Szz
Figure 3.10: The e�ect of the draw ratio E on dimensionless axial stress.
of the Kelvin strain shows a linear relationship on the draw ratio as displayed in
Figure 3.13. The birefringence measurement also shows an almost linear relation-
ship in Figure 2.11. This suggests that the Kelvin strain can e�ectively represent
the anisotropic structural change during the drawing process. Figure 3.14 shows the
e�ect of the dimensionless drawing force on the Kelvin strain.
3.6 The E�ect of Draw Temperature
The e�ect of drawing temperature (Tmax) is examined by choosing Tmax as 1100C,
1150C, 1200C and 1215C to match the experimental parameters. The feed and
winder speeds are held constant such that wi = 0:048 � 10�3m=s and the draw
ratio is E = 4100. Figure 3.15 shows the dimensionless velocity pro�les showing
57
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 16
5
4
3
2
−10
−10
−10
−10
−10
−101
Z
Rad
ial S
tres
s
E=587
E=1150
E=3200
E=4410
Srr
Figure 3.11: The e�ect of the draw ratio E on dimensionless radial stress.
58
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z
Rad
ius
E=587
E=1150
E=3200
E=4410
R
Figure 3.12: The e�ect of the draw ratio E on dimensionless radius.
59
0
2
4
6
8x 10
−5
Tm=1150C →
Tm=1215C →
500 1000 1500 2000 2500 3000 3500 4000 4500
Draw Ratio
Kel
vin
Str
ain
ε 1
Figure 3.13: The e�ect of the draw ratio E and draw temperature Tmax on the Kelvinstrain.
60
90 95 100 105 110 115 120 1250
0.5
1
1.5
2
2.5
3x 10
−5
Dimensionless Drawing Force
Kel
vin
Str
ain
Figure 3.14: The e�ect of the dimensionless drawing force on Kelvin strain at Tmax =1215C.
61
changing slopes for various drawing temperatures. The absolute values of the axial
and radial stresses increase with the decreasing drawing temperature as shown in
Figures 3.16 and 3.17. As drawing temperature decreases the viscosity increases,
resulting in increasing drawing force. Figure 3.18 displays the �ber radius change for
various drawing temperatures. Low drawing temperature causes a sharp neck-down
convergence of the �ber radius. Fiber birefringence (Kelvin strain) decreases with
the increase of the drawing temperature shown in Figure 3.19. Figure 3.19 also shows
the e�ect of the cooling rate on �ber birefringence. The drawing stress increases with
the decrease of Tmax and so does the birefringence. Thus increasing drawing stress
results in larger �ber birefringence. If the Kelvin strain in Figure 3.19 is scaled by
the each of the feed speed wi, the two curves overlap and become a universal curve
shown in Figure 3.20.
3.7 Conclusion
A simple Je�rey model has been used as the constitutive equation, combined
with the mass, momentum and energy equations to model the glass �ber drawing
process. The Je�rey model simulates not only the overall drawing process, but also
the elastic strain corresponding to �ber birefringence. The model results show good
agreement with the experimentally determined birefringence data shown in Figure
2.13. The Kelvin strain has a linear relationship with the draw ratio. As the drawing
temperature increases, the Kelvin strain decreases. The viscoelastic behavior has lit-
tle e�ect on the kinematics and dynamics for high temperature during �ber drawing.
Under the same drawing conditions, the viscoelastic simulation (Maxwell and Je�rey
model) displayed little di�erence from that of the Newtonian ow for the drawing
variables such as velocity, axial stress, radial stress and the radius pro�le along the
62
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
500
1000
1500
2000
2500
3000
3500
4000
4500
Z
Vel
ocity
Increasing temperature
w
Figure 3.15: The e�ect of the draw temperature Tmax on dimensionless velocity, E =4410. Tmax=1100, 1150, 1200 and 1215C
63
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.5 1
1.5 2
2.5 3
3.5 4x 10
5
Z
Axial Stress Szz
Figu
re3.16:
Thee�ect
ofthedraw
temperatu
reTmaxon
axial
stress,E
=4410.
Tmax =
1100,1150,
1200and1215C
64
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1−
2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0
x 105
Z
Radial Stress Srr
Figu
re3.17:
Thee�ect
ofthedraw
temperatu
reTmaxon
radiusstress,
E=
4410.Tmax =
1100,1150,
1200and1215C
65
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z
Rad
ius
Increasing temperature
R
Figure 3.18: The e�ect of the drawing temperature Tmax on neck-down region, E =4410. Tmax=1100, 1150, 1200 and 1215C
66
1100 1120 1140 1160 1180 1200 12200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6x 10
−4
← wi=0.027x10 m/s
← wi=0.048x10 m/s
Drawing Temperature in C
Kel
vin
Str
ain
ε 1
-3
-3
Figure 3.19: The e�ect of the draw temperature Tmax and feed speed on Kelvinstrain, E = 4410.
67
1100 1120 1140 1160 1180 1200 12200.5
1
1.5
2
2.5
3
3.5
Drawing Temperature in C
Sca
led
Kel
vin
Str
ain
ε 1
Figure 3.20: The e�ect of the draw temperature Tmax and feed speed on scaled Kelvinstrain, E = 4410.
68
spinline. Hence, the viscoelastic e�ects can be determined after the kinematic are
obtained.
The large deformation during the drawing process occurs in the high temperature
range where the glass viscosity is very low and the ow is nearly Newtonian. The
simulation shows that all variables change mainly in the high temperature range.
Saturation of the Kelvin strain at high temperature occurs because the retardation
time is small compared to residence time during deformation at these higher tem-
peratures. As the �ber is rapidly cooled the variables stop changing and the state is
\frozen". However, viscous Newtonian ow and the viscoelastic Maxwell model are
unable to model the anisotropic structure change in the drawing process, and only
the Je�rey model is capable of simulating the frozen elastic strain and its relaxation
process successfully.
CHAPTER IV
RELAXATION MODELING
4.1 Introduction
The relaxation of �ber birefringence during annealing is investigated. Glass �ber
birefringence is the optically accessible manifestation of anisotropic structure that
relaxes in a time-and temperature-dependent manner. Experiments in this chapter
show that this relaxation near the glass transition range is similar to that of viscoelas-
tic strain relaxation, however at lower temperature annealing, �ber birefringence does
not fully relax.
Structural relaxation is the process by which material thermodynamic properties,
such as enthalpy, volume, and refractive index, gradually approach their equilibrium
values following changes in some external parameters, such as temperature and pres-
sure. In general, structural relaxation is described by
P (T; t)� P (T;1)
P (T; 0)� P (T;1)= M(T; t); (4.1)
where P (T; t) is any thermodynamic property of temperature T and time t, P (T;1)
is its equilibrium value, P (T; 0) is the initial value, M(T; t) is a relaxation function
that for an isothermal process may have the form
M(T; t) = e�t=�(T ); (4.2)
69
70
For oxide glasses in the annealing range, the �ctive temperature (the temperature
at which the present structure would be in equilibrium [46]) is commonly used to
describe structural relaxation such that
dTfdt
=T � Tf
�(4.3)
where the time constant � has the linearized Arrhenius dependence on temperature
and the �ctive temperature as
� = �0 exp(�A1T � A2Tf); (4.4)
where �0, A1, and A2 are constants. Tool's approach successfully captures the thermal
expansion of annealed glass by using
M = �g(T � T0)� �(Tf � Tf0); (4.5)
where �g is the glassy state thermal expansion coe�cient, T0 is a reference temper-
ature, � is a structural coe�cient, and Tf0 is the initial �ctive temperature. This
approach was unable to predict the thermal expansion data of quenched glass under
500C. Refractive index relaxation experiments by Ritland [47] have shown that a
single �ctive temperature evolution described by (4.3) and (4.4) is not su�cient to
describe the distributed nature of property relaxation. Since the as-drawn �ber is
anisotropic, it is not in equilibrium for any temperature and we have found that
only partial relaxation may occur, the use of a single �ctive temperature to describe
drawn �ber birefringence relaxation is questionable.
Narayanaswamy [48] proposed using a reduced time � to partition the temperature
and �ctive temperature e�ects. Linearity can be restored by using the reduced time
parameter de�ned by
� = �(T0)Z t
0
dt0
�; (4.6)
71
where T0 is an arbitrary reference temperature. Boltzmann's superposition principle
describes Tf by
Tf = T �Z �
0M(� � �0)
dT
d�0d�0: (4.7)
The thermodynamic property P (T; �) becomes
P (T; �) = P (T;1)� �s
Z �
0M(� � �0)
dT
d�0d�0; (4.8)
where M(� � �0) is a relaxation function most often described by a \stretched expo-
nential"
M(�) = e�(�=�)b
(4.9)
for a constant b, and �s is a structure parameter.
Based on the general Arrhenius relation � = �0 exp(4HRT
), to describe thermally
activated relaxation, Narayanaswamy suggested that the contribution of �ctive tem-
perature to structural relaxation could be included by partitioning the relaxation
time in the following empirical form
� = �0 exp
x4HRT
+(1� x)4H
RTf
!; (4.10)
where �0 is a constant, 4H is an activation energy, R is the ideal gas constant, and x
is a constant between 0 and 1. Later investigators [49, 50, 51] used the Adams-Gibbs
equation based on the suggestion that the ow of the structure involves the coop-
erative rearrangement of increasingly larger numbers of molecules as temperature
decreases. Thus the relaxation time depends on the con�gurational entropy as
� = �0 exp
A
T4S(Tk; Tf)
!; (4.11)
where 4S(Tk; Tf) is the con�gurational entropy with structure corresponding to Tf ,
the Kauzmann temperature Tk is de�ned by the vanishing of the equilibrium value of
72
4S, and �0 and A are constants. The phenomenological models based on the Tool-
Narayanaswamy (T-N) equations successfully describe relaxation of those isotropic
parameters such as density, refractive index and enthalpy in bulk glass in the glass
transition range. Recently, di�erential scanning calorimetry (DSC) experiments in
oxide glass �bers by Huang and Gupta [52] demonstrate that the T-N model is inad-
equate for describing enthalpy relaxation of drawn glass �bers at temperatures well
below the glass transition range. Relaxation well below the glass transition temper-
ature has also been observed in oxide glass [53], metal glass [54], and polymers [55].
Since birefringence in glass �bers is the result of frozen anisotropic elastic strain,
its relaxation is expected to behave similarly to that of viscoelastic strain recovery
of inorganic glasses. The experiments in this study demonstrate this behavior for
birefringence relaxation near the glass transition temperature Tg. The use of a re-
laxation spectrum to simulate the relaxation of glass �ber birefringence is explored.
Low temperature (sub { Tg) birefringence relaxes to a non{zero value, suggesting
that part of the frozen elastic strain remains. Over a very broad temperature range,
the relaxation process is not only nonlinear but also non{thermal rheologically sim-
ple, i.e. the form of the relaxation function changes with temperature. The stretched
exponential function is modi�ed to simulate the birefringence relaxation over a broad
range of annealing temperatures. A temperature dependent parameter is introduced
in the stretched exponential function to account for the unrelaxed part of the frozen
elastic strain. The Je�rey model used in the �ber drawing process is also explored
to model the birefringence relaxation. The Je�rey model has a relaxation time and
retardation time thus it contains the two characteristic times needed to capture the
relaxed and partially relaxed responses.
73
4.2 Stretched Exponential Modeling
Our experimental results show that glass �ber birefringence relaxation in the
glass transition range is qualitatively similar to that of the viscoelastic strain recov-
ery of oxide glasses [56] in the glass transition temperature range. Glass structure
and viscoelastic strain relaxation have been modeled using the stretched exponential
(sometimes called the Kohlrausch-Williams-Watts or KWW function).
M(t) = e�(t=�)b
(4.12)
where M(t) is the relaxation function, � is the relaxation time, and b is a constant.
Previous investigations have shown that b is around 0.5 for oxide glass stress relax-
ation or viscoelastic strain relaxation [43] in the glass transition range. We explore
the ability of the stretched exponential to simulate the glass �ber birefringence re-
laxation with the typical value of b = 0:5. The results for the scaled birefringence
relaxation are shown in Figure 4.1 using the relaxation times shown in the caption,
chosen to best �t the data. Here, � is assumed to be constant for the annealing time,
so that the heating and cooling times of the �ber are neglected.
We can compare these relaxation times to those determined from viscosity and
the elastic modulus as
� =�
G;
where � is viscosity and G is the shear modulus of glass (assumed constant). As
in Gupta [37], we �nd that the WLF and Arrhenius relationships are inadequate
to describe the viscosity over a very large temperature range. Instead we use the
Walther double exponential correlation
� = �0 exp[exp(v0 � v1 ln �)];
74
0 20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (min)
Sca
led
Bire
frin
genc
e
(i)
(ii)
(iii)
(iv)
(v)
Figure 4.1: Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (13) with b = 0:5, and (i) T = 309C,� = 1000min; (ii) T = 360C, � = 150min; (iii) T = 387C, � = 50min;(iv) T = 408C, � = 20min; (v) T = 511C, � = 5min.
where �0, v0 and v1 are constants for varying absolute temperature T . The correlation
coe�cients for borosilicate glass are �0 = 3:0 Pa� s, v0 = 18 and v1 = 2:18 lnK�1 from
viscosity versus temperature data of Dormeus [3]. Using the Walther correlation
results in a ratio of time constants of 1016 between T = 309C and T = 511C rather
than the factor of 200 shown in Figure 4.1. Clearly, the relaxation phenomenon is
more complicated than what can be related by a single exponential relaxation time
and the glass cannot be viewed as thermo-rheologically simple.
75
As Figure 4.1 shows at high annealing temperatures (T = 408C and 511C) the
stretched exponential with b = 0:5 does not �t the data well as the birefringence
appears to relax to zero in �nite time, while the simulation over-predicts the time of
e�ectively complete relaxation by a factor of �ve. At low temperatures (T = 309C)
the birefringence does not fully relax for long times (24 hours in Figure 2.15). The
exponential constant b may be varied for di�erent relaxation temperatures without
solving the problem.
Similarly, one relaxation case is modeled using Tf and (4.3), (4.4), (4.6), and
(4.9) with b = 1. For this case, we use the experimentally-determined time constant
of 14 seconds for the temperature rise of the �bers at the beginning of the annealing
process. Tf0 = 1273K is assumed to correspond with temperatures slightly lower
than Tm, where most deformation will occur. The �ctive and �ber temperatures are
shown in Figure 4.2a for �0 = 1027s, a1 = 5�10�2K�1, and a2 = 2:3�10�2K�1 taken
from Tool [46]. The evolution of the relaxation time � and the scaled birefringence are
then shown on Figures 4.2b and 4.2c, respectively. As expected, this time-marching
simulation shows that the scaled birefringence decays to zero, and now, the relaxation
is slow at initial times. Clearly, this approach does not model the observed relaxation
either.
76
0 50 100 150 200200
400
600
800
1000
1200
1400
Time (s)0 50 100 150 200
100
102
104
106
108
Time (s)τ
(s)
0 50 100 150 20010
−2
10−1
100
Time (s)
Sca
led
Bire
frin
ge
nce
T &
T (K
)f
(i)
(ii)
(a) (b)
(c)
Figure 4.2: Fictive temperature modeling of relaxation. (a) Temperature variationduring annealing: (i) Fictive temperature Tf and (ii) �ber temperatureT ; (b) Relaxation time evolution; and (c) Birefringence relaxation.
77
To capture the relaxation at low temperatures the stretched exponential is mod-
i�ed by introducing a temperature dependent constant c(T ) such that
M(t) = [1� c(T )]e�(t=�)b
+ c(T ) (4.13)
Figure 4.3 shows the simulation using (4.13) in a simpli�ed form using b = 1. This
modi�ed relaxation function is capable of capturing all the relaxation data in Figure
2.14.
4.3 Je�rey Modeling
Equation (4.13) shows that relaxation can be modeled using two exponential
elements with at least one of them \frozen" at low annealing temperatures (i.e.
with very large relaxation times) to capture the incomplete relaxation. Equation
(4.13) o�ers limited physical interpretation and cannot simulate the development
of birefringence during the drawing process. A simple, two-component linearized
Je�rey element model is explored, shown in Figure 4.4, to simulate both the drawing
and relaxation processes. Here, either a tensile stress � or elongation rate _� can be
applied.
In the parallel Je�rey model, elements �12 and �22 represent the large viscous
deformation produced during the drawing process at high temperature. The two
Kelvin elements represent the anisotropic elastic strain that is frozen during �ber
cooling and relaxed during annealing. A minimum of two elements (two relaxation
times) is required for incomplete relaxation. Since birefringence is a measure of
anisotropic strain, we assume birefringence is proportional to the total strain in the
two Kelvin elements. This anisotropic strain may partially or fully relax depending
on the annealing temperature. The relaxation rate and extent of relaxation depend
78
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (min)0 20 40 60 80 100 120 140 160 180
Sca
led
Bire
frin
genc
e
(i)
(ii)
(iii)
(iv)(v)
Figure 4.3: Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using (4.13) with b = 1, and (i) T = 309C,� = 50min, c(T ) = 0:65; (ii) T = 360C, � = 47min, c(T ) = 0:35; (iii)T = 387C, � = 45min, c(T ) = 0:05; (iv) T = 408C, � = 16:7min,c(T ) = 0:0; (v) T = 511C, � = 8:24min, c(T ) = 0:0.
79
µ11
µ12
µ21
µ22
G1
G 2
σ or ε
Figure 4.4: Two Je�rey elements in parallel
on the temperature dependent viscosities assigned to the viscous elements. The
moduli of the spring elements are assumed constant.
Birefringence relaxation simulations are conducted at various annealing temper-
atures using the quenched system after �ber drawing. Thermal expansion e�ects are
neglected. The simplest temperature versus relaxation study using the 2-element
Je�rey model varies one parameter with temperature. The two elastic elements are
assumed to have a constant modulus of 24.4GPa. Here for simplicity, three of the vis-
cous elements are assumed to have no temperature dependence in the annealing tem-
perature range and their values are held as �21 = 1:1�1013Pa�s, �12 = 2:2�1011Pa�s,
�22 = 2:2 � 109Pa�s. The viscosity of the �nal viscous element �11 varies with an-
nealing temperatures as shown in Figure 4.5. This simple simulation of the parallel
Je�rey model is shown in Figure 4.6 to model the birefringence relaxation of glass
�bers over the entire range of annealing temperatures.
4.4 Conclusion
The generally used stretched exponential for delayed elastic strain and struc-
tural relaxation is not able to accurately simulate observed birefringence relaxation.
80
300 320 340 360 380 400 420 440 460 480 50010
10
10
10
10
13
14
15
16
17
Temperature (C)
Vis
cosi
ty (
Pa
s)
Figure 4.5: �11 versus annealing temperatures.
81
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time(min)
Sca
led
Bire
frin
genc
e
Figure 4.6: Birefringence relaxation for �bers drawn at Tm = 1215C. Data fromFigure 2.14. Lines are plotted using the two Je�rey elements in parallelmodel.
82
A modi�ed exponential model with incomplete relaxation simulates relaxation satis-
factorily with an empirical temperature dependent constant. A parallel Je�rey model
with temperature dependent coe�cients simulates the glass �ber birefringence relax-
ation well not only for the high temperature annealing but also for low temperature
annealing.
The Je�rey model enables simulation of the development of birefringence dur-
ing �ber drawing as well as the relaxation process. The various elements in the
Je�rey model lend themselves to interpretation as viscous deformation elements or
viscoelastic elements storing anisotropic strain at low temperature. The complete
and incomplete relaxation behavior requires two temperature dependent relaxation
times. The empirical constant in the modi�ed exponential model represents the sec-
ond branch of the parallel Je�rey model and e�ectively provides a second retardation
time for modeling complete and incomplete relaxation.
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
Our experiments and modeling show that optical anisotropy develops during �ber
drawing in glass �bers drawn from preforms. Optical anisotropy is the manifestation
of the structural anisotropy that develops during the drawing process. In oxide
glasses the structural and optical anisotropies are consequences of frozen anisotropic
elastic strain by rapid cooling under the drawing load. The anisotropy of glass
�bers has been determined from birefringence measurements for various drawing
parameters.
The amount of the anisotropy in as-drawn glass �bers is shown to be strongly
a�ected by the draw temperature, the draw ratio and the draw rate. Fiber bire-
fringence increases with decreasing drawing temperature, increasing draw ratio, and
increasing cooling rate. Increasing temperature decreases the glass melt viscosity
and hence the drawing force decreases, and �ber birefringence decreases accord-
ingly. Rapid cooling halts relaxation after the axial force is no longer applied. Thus
the structure is quenched in an anisotropically strained state and the anisotropic
structure exhibits birefringence. Increasing drawing speed increases draw ratio, this
increased deformation also results in increased structural anisotropy manifested as
birefringence.
83
84
A simple Je�rey model has been used as the constitutive equation for glass �ber
drawing, and combined with the mass, momentum and energy conservation equations
to model the glass �ber drawing process. The Je�rey model is able to simulate the
overall drawing process, and stored anisotropic elastic strain energy associated with
�ber birefringence.
The viscoelastic parameters have little e�ect on the kinematics and dynamics
of �ber drawing. Under the same drawing conditions viscoelastic simulations via
the Maxwell and Je�rey models displayed little di�erence from that of a Newtonian
model in drawing variables such as velocity, axial stress, radial stress and the radius
pro�le along the spinline. This is true because the majority of the large deformation
during drawing occurs at high temperatures wherein the glass viscosity is very low
and Newtonian viscous ow dominates. The �ber drawing simulations show that the
kinematic and dynamic variables change dramatically in the high temperature range.
Saturation of the Kelvin strain at high temperature occurs because retardation time
is small compared to residence-time during deformation at these high temperatures.
The �ber is cooled rapidly through the glass transition to the glassy (elastic solid)
state. However, the Newtonian model and the viscoelastic Maxwell model are unable
to store the anisotropic structure change during the drawing process, and the Je�rey
model is the simplest viscoelastic model capable of simulating frozen elastic strain.
The Je�rey model is used to predict the change in as-drawn �ber birefringence with
drawing temperature and draw ratio.
Glass �ber birefringence relaxation at temperatures well below the glass transi-
tion range in oxide glasses has been observed. At low temperatures, birefringence
relaxation is observed, but relaxation is incomplete. This temperature dependent
relaxation behavior suggests a broad spectrum of relaxation times. The generally
85
used stretched exponential for delayed elastic strain and structural relaxation is not
able to accurately simulate observed birefringence relaxation. An ad hoc modi�ed
exponential model without a stretching factor but with an empirically determined
temperature dependent constant models incomplete relaxation satisfactorily. A par-
allel Je�rey model with temperature dependent coe�cients simulates the glass �ber
birefringence relaxation well not only for high temperature annealing resulting in
complete relaxation but also for low temperature annealing resulting in incomplete
relaxation. The Je�rey model enables simulation of the development of birefringence
during �ber drawing as well as the relaxation process. The various elements in the
Je�rey model lend themselves to interpretation as viscous deformation elements or
viscoelastic elements storing anisotropic strain at low temperature.
The single Je�rey model is the simplest constitutive model capable of captur-
ing the kinematic and dynamic parameters during �ber drawing and the as-drawn
�ber birefringence. The complete and incomplete relaxation behavior requires two
temperature dependent relaxation times. The empirical constant in the modi�ed
exponential model represents the second branch of the parallel Je�rey model and
e�ectively provides a second retardation time for modeling complete and incomplete
relaxation.
Although it is shown that both glass �ber tensile strength and birefringence are
a�ected by the drawing process parameters, the correlation between these two prop-
erties is still not very clear. More work should be done to further study the correlation
between the anisotropic structural change and the �ber mechanical properties. It is
also important to understand how the anisotropic structural change induced during
the drawing process a�ects the optical �ber properties.
APPENDICES
86
87
APPENDIX A
CONSTITUTIVE THEORIES OF GLASS NEAR
THE GLASS TRANSITION RANGE
Inorganic glass melts are generally treated as Newtonian uids. However, at tem-
peratures around the glass transition, non-Newtonian behavior in inorganic glasses
has been reported for various glasses and by several authors [4, 5, 57, 58]. The steady-
state stress versus strain rate of a soda-lime-silica glass at T=596C (Tg = 570C) in
Figure A.1 [4] shows deviation from Newtonian behavior at high strain rates. The
transient stress versus strain response at constant strain rate contains a strain rate
and temperature dependent stress overshoot. The strain rate dependent uniaxial
stress-strain response of a silicate glass near the transition range is shown schemati-
cally in Figure A.2. At a low constant applied strain rate the stress rises monotoni-
cally to a steady state value. As the applied constant strain rate ( _�) is increased under
isothermal conditions, the stress reaches a maximum before it softens to a steady-
state value. The magnitudes of both the peak and steady-state stresses increase with
the applied strain rate.
Many previous studies of the rate dependent deformation response of inorganic
glasses have been concerned with the shear thinning behavior, i.e. the non-Newtonian
steady state behavior. Phenomenological models [4, 59, 60] have been developed to
88
00.5
11.5
22.5
33.5
44.5
50 2 4 6 8 10 12 14 16 18
Strain R
ate (10−
4/sec)
Steady−state Stress (10 9dyne/cm2)
Figu
reA.1:
Variation
ofstead
y-state
stressvs.strain
ratefor
asoda-lim
e-silicaglass
atT=596C
show
ingdeviation
fromNew
tonian
behavior
athigh
strainrates.
89
ε
Strain
Str
ess
Figure A.2: Schematic representation of the uniaxial stress vs. strain response ofinorganic glass near Tg at various constant applied strain rates.
90
describe the steady-state stress (or apparent viscosity) versus strain rate. Few at-
tempts have been made to describe the transient stress response, i.e. the stress
overshoot at high strain rates. In practice, high deformation rate processes are en-
countered in many situations, such as in �ber drawing. The �nal �ber properties such
as sti�ness, strength and optical anisotropy depend on the processing temperatures
and strain rates of the glass as it passes through its viscoelastic temperature range.
To fully characterize glass �ber drawing, it is important to consider the transient
deformation response of glass near Tg.
The strain rate or stress dependent non-Newtonian behavior of inorganic glasses
we seek to describe has been observed experimentally in various deformation states [4,
5, 57, 58]. Some authors [4] interpret these responses by comparing to molecular dy-
namic (MD) simulations on a Lennard-Jones potential glass. Both inorganic glass
experiments and the MD simulations show similar behaviors. The MD model indi-
cates that the material develops a layered structure under high shear strain rates
and that causes the drop in the true stress response.
A macroscopic constitutive description of both the transient and the steady state
responses of inorganic glasses near the transient region is explored, in which rate
and temperature dependent stress overshoot and shear thinning in the steady state
response are captured. In this appendix several constitutive modeling approaches are
examined for their ability to predict the rate and temperature dependent response
of glass near Tg in tension and compression.
Rekhson [61] proposed a model for non-linear viscoelastic relaxation based on the
hypothesis of stress-induced structural relaxation (an idea analogous to temperature-
induced relaxation) and introduced the concept of the �ctive stress based on the
phenomenological Tool's equation (replacing the �ctive temperature Tf in Equation
91
(4.3) with �ctive stress), to capture both the transient and the steady state stress.
The model is based on a spectrum of simple viscoelastic relaxations and has been
applied to the uniaxial tensile response of a silicate glass in comparison with experi-
mental data from Rekhson [4]. The relaxation times are functions of both stress and
�ctive stress. This model is phenomenological in that it assumes a �ctive stress and
incorporates a temperature e�ect by correcting the viscosity for di�erent tempera-
tures.
Argon [62] has proposed a model for the resistance of glass to inelastic deformation
that considers the rate and temperature dependent resistance to local transforma-
tions of the atoms in metallic and oxide glasses at low temperatures. This model pre-
dicts the rate and temperature dependent stress maximum in soda glass and E-glass.
Argon and Kuo [63] have studied the mechanism of plastic ow of atomic glasses by
using a disordered soap bubble raft. Their study revealed that under shear the rafts
are observed to change shape by a collection of very local shear transformations. The
2-D computer molecular dynamics approach by Deng and Argon and coworkers [64]
demonstrates that the principal mechanism of plastic strain is local shear transfor-
mations nucleated preferentially in the boundaries of liquid-like material separating
the small quasi-ordered domains. The sites of the maximum shear strain correlate
well with the sites of excess free volume, i.e. the fraction of matter having a lower
atomic coordination. The original rate and temperature dependent model of inelas-
tic deformation resistance for glassy materials proposed by Argon [62] serves as a
useful practical description of the actual deformation mechanism. A description of
the structural state in inorganic glass is examined based on the conceptualizations of
Argon and others [64] of a distribution of free volume within volume elements on the
order of several atomic distances. This distribution evolves with inelastic deforma-
92
tion in a manner that is rate and temperature dependent and results in an overshoot
in the transient stress versus strain response of inorganic glass.
The homogeneous response of glassy polymers also exhibits true stress softening
and strain rate and temperature dependent behavior. A number of investigators
have described this phenomenon near or below the glass transition temperature us-
ing phenomenological macroscopic, mesoscopic, or molecular models [65, 66, 67]. It
is clear that during deformation some structural rearrangement occurs in the form
of local rearrangements of the number and size of density variations in the polymer
microstructure. A micromechanically based constitutive model of this structural evo-
lution process has successfully predicted the rate dependent stress softening response
of glassy polymers [65, 66, 67].
Viscoelastic models such as the generalized Maxwell model are generally used to
describe the behavior of inorganic glasses in the transition range [68]. The White-
Metzner model is a viscoelastic model that involves rate dependent viscosity and has
been used to describe glass �ber drawing [37]. It is similar to the Je�rey model, and
at constant strain rate is equivalent to the Je�rey model. Structure related models
have rarely been explored to describe inorganic glass behavior in the transition range.
The Marrucci model [69] has demonstrated the ability to capture the stress overshoot
in polymers for one dimensional elongational ow. A elastic-viscoplastic model [62]
based on the local thermal transformation theory is examined. This approach is
modeled for tensile and compression tests and compared with the experimental data
to see if it can describe the strain rate and temperature dependent response in the
transient and steady state ranges.
93
Figure A.3: Schematic two-dimensional representation of the structure of a hypo-thetical compound A2O3
A.1 A Viscoplastic Theory
A.1.1 Physical Description of Oxide Glasses
The deformation of inorganic glass is closely related to its structure. Zachariasen's
random network hypothesis [2] is largely accepted as the classical theory to describe
the structure and properties of unmodi�ed oxide glasses in the solid state. Figure A.3
is a schematic 2D diagram showing the structure of a hypothetical oxide A2O3 in
the glassy form [2]. In this material the basic polyhedron is the AO3 triangle. The
triangles are joined only at their corners by bridging oxygens. In silica glass, the
basic polyhedron is the SiO4 tetrahedron.
For modi�ed oxide glasses, the introduction of a modi�er such as Na2O will
change the random network structure in which the alkali and alkaline earth ions are
94
Figure A.4: Two-dimensional representation of the structure of a modi�ed oxide glass
incorporated into the structure shown in Figure A.4 [2]. The cations C are situated
in the relatively large voids between the silica bonds and for each additional oxygen
anion introduced, one A-O-A bridge is broken so that two non-bridging oxygens are
formed.
Glass and glass forming melts are considered to have a large number of free volume
sites. Although glasses at low temperatures have little plastic ow before rupture
failure, it is expected that plastic ow prevails at the glass transition range. As in
the structure shown in Figure A.4, the modi�ers introduced into the glass create
larger free volume sites. 2-D MD simulations and soap bubble experiments [63, 64]
have shown that the free volume sites are the primary local transformation sites for
the mechanism of plastic deformation. These free volume sites can be characterized
as free energy barrier sites [62], and large free volume sites correspond to low free
95
energy barrier sites. The transformation of a series of bonded atoms from one con-
formation to another has to overcome the local free energy barrier. According to the
transition state theory, local rearrangements are thermally activated, and the local
transformation rate, !, can be expressed as an Arrhenius form
! = !0exp��F
kT
�; (A.1)
where, !0 is the attempt frequency; �F is the activation free energy for the trans-
formation; k is the Boltzmann constant and T is the absolute temperature. Under
an externally applied stress, the activation free energy is modi�ed as [70]
�F = �F0
�1�
��
�0
�p�q; (A.2)
where �F0 is the activation energy without external stress; � is the local e�ective
stress, �0 is the limiting stress, and p and q are material dependent parameters.
According to the Argon model which describes the transformation extent of inorganic
glasses as an exponential form of the free enthalpy change [71], the total inelastic
strain rate is given by
_ = _ 0exp���F0
kT
�1�
��
�0
�p�q�; (A.3)
where _ is the inelastic strain rate, and _ 0 is the pre-exponential factor determined
experimentally. Equation (A.3) can also be written in the following form [65]
_ = _ 0exp��AsT
�1�
��
s
�p�q�; (A.4)
In equation (A.4), s is the athermal shear strength of the material and it is a structure
related parameter. The quantity As is the zero stress level activation energy, and � is
96
unsheared sheared
Figure A.5: Schematic illustration of glass melts under shear
the e�ective ow stress. This inelastic deformation process is rate and temperature
dependent. We assume the athermal shear strength is evolving during plastic defor-
mation processes to a steady state sss as the structure becomes layered as shown in
Figure A.5. The proposed evolution law for s is given in equation (A.5) [72].
_s = h(1� s
sss) _ ; (A.5)
where h is the evolution slope. sss is given as sss = as0, here s0 =0:77�1��
[66], � is the
elastic shear modulus, and a is a rate and temperature dependent parameter. � is
the Poisson's ratio.
A.1.2 The Constitutive Formation
Various experiments show that the transient response of inorganic glasses is char-
acterized as initially elastic until yield occurs at a stress which depends upon tem-
perature and strain rate, and then plastic ow ensues. Therefore we seek an elastic-
viscoplastic constitutive framework for describing the response. The kinematics for
the �nite deformation are implemented as follows. The velocity gradient tensor L
97
can be expressed as
L = D+W; (A.6)
L = Le + FeLpFe�1; (A.7)
where D is the symmetric part of L, and W is the skew-symmetric part of L. Fe
is the elastic deformation gradient, Fe�1 is its inverse and Lp is the plastic velocity
gradient. For the one dimensional case of uniaxial ow, equation (A.7) has the form
L11 = Le11 + Lp
11; (A.8)
where L11 is the total velocity gradient, or the total strain rate _�. The elastic velocity
gradient Le11 can be written as Le
11 = _�e, where _�e represents the elastic strain rate.
If linear elastic behavior is assumed, �e = �E, where � is the applied axial stress for
one dimensional uniaxial ow and E is the Young's modulus. With equation (A.4)
for the inelastic strain rate where _ p =p3_�p and �e = �
E, equation (A.8) becomes
_� =_�
E+
_ 0p3exp
�AsT
1�
�p3s
!p!q!; (A.9)
A.1.3 Model Prediction for Elongational Tests
Previous experiments examining the inorganic glass deformation response have
been documented in the literature [4, 5, 57, 58]. The results show behavior similar
to that displayed in Figure A.2. Uniaxial tension and compression data from [4, 5]
will be used to compare with the model simulations.
Figures A.6 and A.7 show the uniaxial tension deformation responses of soda-lime-
silicate glass for two di�erent temperatures taken from [4]. The apparent viscosity
98
0
50
100
150
200
250
300
0 100 200 300 400 500 600 700 800
Str
ess
(MP
a)
Time (s)
Figure A.6: Stress vs. time response of soda-lime silicate glass tension tests at variousstrain rates at T=836K.
versus time graphs of [4] are transformed into uniaxial stress versus time using � =
3� _� in order to compare with the simulations of equation (A.9).
The material parameters in equation (A.9) are determined as follows. For inor-
ganic glasses, Eyring's approach is used, i.e. p and q are taken as q = 1 and p = 1
[67]. _ 0 and A are determined from the data in Figures A.6 using maximum stress
versus strain rate data pairs and equation (A.4). The elastic parameters, Young's
modulus for soda-lime silicate glass at elevated temperatures, have been measured
by McDraw [73]. Here we take the Young's modulus at a temperature of around
600C. The parameters needed for the computation are listed in Table A.1.
Figure A.8 shows the results of the simulation and comparison with the exper-
99
0
50
100
150
200
250
300
0 50 100 150 200
Str
ess
(MP
a)
Time (s)
Figure A.7: Stress vs. time response of soda-lime silicate glass tension tests at variousstrain rates at T=866K.
Table A.1: Parameters used in tension simulations
_ 0(1/s) A(K/MPa) E(MPa) � a h(MPa/s)
T=836K 1.1E+5 22.61 20,000 0.21 0.97-0.999 31,500
T=866K 1.1E+5 22.61 20,000 0.21 0.97-0.999 31,500
100
0
50
100
150
200
250
300
350
0 0.05 0.1 0.15 0.2 0.25 0.3
Str
ess
(MP
a)
Strain
ε= 0.0014 /sec
ε= 0.00072 /sec
ε= 0.00036 /sec
ε= 0.00018 /sec
ε= 0.00009 /sec
ε= 0.000036 /sec
Figure A.8: Stress vs. time response of soda-lime silicate glass for tension tests atvarious strain rates at T=836K. Curves are model simulations using(A.9), and symbols are the data from [4].
101
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Str
ess
(MP
a)
Strain
ε=0.0181 /sec
ε=0.0072 /sec
ε=0.0036 /sec
ε=0.0018 /sec
ε=0.00091 /sec
Figure A.9: Stress vs. time response of soda-lime silicate glass for tension tests atvarious strain rates at T=866K. Curves are model simulations using(A.9), and symbols are the data from [4].
102
Table A.2: Parameters used in compression simulations
_ 0(1/s) A(K/MPa) E(MPa) � m0 h(MPa/s)
T=869K 1232.96 13.16 20,580 0.21 6:58� 10�3 5000
T=902K 6.88 13.16 9800 0.21 6:58� 10�3 5000
T=930K 27.45 28.12 4900 0.21 6:58� 10�3 5000
imental data at T=836K. The curves at _�=0.0072/sec, 0.0036/sec and 0.0018/sec
were used to obtain the parameters in Table A.1, and the remaining curves are pre-
dictions. It is shown that the quality of the �t and the predictive capability are
quite good for the strain rate range 0.000036-0.0072/sec. The model overpredicts
the stress overshoot at 0.0014/sec. The simulations at T=866K are shown in Figure
A.9 in which all curves are predictions. The theory captures the data fairly well for
the three highest rates; the transient and steady state responses are overpredicted
at the lower rates.
A.1.4 Model Prediction for Compression Tests
Equation (A.9) is also used to simulate the uniaxial compression responses at
constant displacement rates for soda-lime-silicate glass at three di�erent tempera-
tures from reference [5]. The relation between strain rate and displacement rate is
_� = _h=h, where h is the current height of the sample.
We have found that the model is not predictive if we use the same approach as in
the tensile test simulation; using the data at one temperature to determine the model
parameters is not predictive of the response at other temperatures. The reason for
this is not clear but the data show increasing rate dependence with decreasing tem-
103
0 0.2 0.4 0.6 0.8 1 1.2 1.40
50
100
150
200
250
300
Str
ess
(MP
a) h=0.022mm/s
h=0.045mm/s
h=0.096mm/s
h=0.067mm/s
h=0.031mm/s
Figure A.10: Stress vs. time response of soda-lime silicate glass for compression testsat various displacement rates at T=869K. [5] and comparison with thepredictions using (A.9).
104
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
50
100
150
200
250
300
350
400
Str
ess
(MP
a) h=3.1mm/s
h=2.2mm/s
h=1.5mm/sh=1.0mm/s
h=0.75mm/s
Strain
Figure A.11: Stress vs. time response of soda-lime silicate glass for compression testsat various displacement rates at T=902K. [5] and comparison with thepredictions using (A.9).
105
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
20
40
60
80
100
120
140
Str
ess
(MP
a)
h=4.0mm/s
h=3.0mm/s
h=2.0mm/s
h=1.0mm/sh=0.7mm/s
Strain
Figure A.12: Stress vs. time response of soda-lime silicate glass for compression testsat various displacement rates at T=930K. [5] and comparison with thepredictions using (A.9).
106
perature, which contradicts modeling assumptions. Thus we have adopted another
approach to attempt to predict the compression data. The compression tests are
carried out in three temperatures as 869K, 902K and 930K. It is found that in this
temperature range the elastic shear modulus is not constant [73], but temperature
dependent. We have used the temperature dependent shear modulus from refer-
ence [73], and have determined other parameters in the model ( _ 0 and A) individually
for various temperatures. In addition, the softening parameter a is also temperature
and displacement rate dependent. The relationship between a and temperature T
and displacement rate _h is given as a = (1�m(T )) exp���
_h0:3063(T�869)+1:2
��+m(T )
by curve �tting, where m(T ) = m0[(T � 869)3 + 27:42(T � 869)2] and m0 is shown
in Table A.2. The model predictions and comparison with the data for three tem-
peratures and various displacement rates are shown in Figures A.10, A.11 and A.12.
The parameters used in the simulation are listed in Table A.2.
Since the compression data are carried out with constant displacement rate in-
stead of constant strain rate, the stresses are expected �rst to decrease after yield
and then to increase with the increasing strain rate during compression. The sim-
ulation successfully captures this feature. However, the elastic-viscoplastic model
does not include viscoelastic behavior before yield which is shown in the experi-
ments, especially at the lower displacement rates. Overall the model successfully
captures the strain rate and temperature dependence of the deformation response of
the glass and the transient behavior (stress overshoot) is also captured by using a
phenomenological evolution structure related parameter model.
107
A.2 Viscoelastic Theories
Various viscoelastic models based on continuum mechanics or molecular struc-
tures are developed to describe the mechanical response of glass and polymer melts.
Two of these models, the White-Metzner model [74] and the Marrucci model [69]
are chosen here to simulate the glass response near Tg and compare with the tension
data at 836K.
A.2.1 White-Metzner Model
The White-Metzner constitutive equation [74] is often used to describe the vis-
coelastic response of polymer and glass melts. The equation for uniaxial ow with a
single relaxation time has the following form.
_�zz +1 + � _��
p3
��zz = 2G _�; (A.10)
_�rr +1 + � _��
p3
��rr = �G _�; (A.11)
where �zz and �rr are the stress components along the axial and radial directions
respectively, _� is the elongational strain rate, � and G are the relaxation time and
the shear modulus of the glass respectively, and � is an adjustable parameter. The
parameters used here are � = 18; � = 18:91; G = 8130MPa to �t the data.
Figure A.13 shows the prediction of the White-Metzner model and comparison
with experimental data in tension at T=836K. The White-Metzner model can not
predict the transient deformation response, i.e. the stress overshoot at high strain
rate, and the rate dependence is also not accurately predicted for the range of strain
rates prescribed.
108
0
50
100
150
200
250
300
350
0 0.05 0.1 0.15 0.2 0.25 0.3
Str
ess
(MP
a)
Strain
= 0.00072 /secε
= 0.00036 /secε
= 0.00009 /secε
= 0.00018 /secε
= 0.000036 /secε
Figure A.13: Stress vs. time of soda-lime silicate glass for various strain rates atT=930K. [5] and comparison with the prediction using (A.10) and(A.11).
109
A.2.2 Marrucci Network Model
Marrucci et al. have proposed a network polymer model [69] in which a structure
related variable is introduced to describe the formation and destruction of the net-
work entanglements. This model has demonstrated the ability to capture the stress
overshoot in polymers for one dimensional elongational ow. We expect that the
deformation of inorganic glasses accompanies structure rearrangement. The defor-
mation response approaches a steady state when these processes reach an equilibrium
state (layered structure). We employ this model to predict the inorganic glass re-
sponse and compare with the data at T=836K. The Marrucci model for elongational
ow with a single relaxation time has the form:
_�zz = x3�zz _x + 2_��zz � 1
�x1:4�zz + 2xG _�; (A.12)
_�rr = x3�rr _x� _��rr � 1
�x1:4�rr � xG _�; (A.13)
_x =x�1:4
�(1� x)� �x�1:4
�(�zz + 2�rr
2Gx)1
2 ; (A.14)
�zz(0) = �rr(0) = 0; x(0) = 1; (A.15)
where x is a structure related variable which is evolving with deformation. All the
other variables are as de�ned before. The parameters are taken as � = 7:0; � =
18:95; G = 8130MPa to �t the data.
The results of simulations using the Marrucci model and comparison with the
tension data for glass at T=836K are shown in Figure A.14.
110
0
50
100
150
200
250
300
350
0 0.05 0.1 0.15 0.2 0.25 0.3
Str
ess
(MP
a)
Strain
= 0.00072 /secε
= 0.00036 /secε
= 0.00009 /secε
= 0.00018 /secε
= 0.000036 /secε
Figure A.14: Stress vs. time response of soda-lime silicate glass in tension for variousstrain rates at T=836K. Curves are Marrucci model simulations using(A.12)-(A.15), and symbols are the data from reference [4].
111
The Marrucci model successfully predicts the stress overshoot at high strain rates
and the reduction extent of the overshoot decreases with decreasing strain rate. How-
ever the strain rate dependence of the steady-state response has not been satisfacto-
rily predicted. Temperature dependence is not included in this model.
A.3 Conclusion
An approach for modeling the mechanical response of inorganic glasses in the
transition range based on a viscoplastic deformation theory has been examined. A
phenomenological model for the structural evolution during deformation has been
proposed to simulate the strain rate and temperature dependent response of glass
near Tg. The simulation of the model and comparison with the uniaxial experimental
data have shown that the model can predict the uniaxial deformation response of
soda-lime-silicate glass around the transition range. The transient stress overshoot
and the rate dependence of the glass are successfully predicted for various temper-
atures for uniaxial tension deformation. In compression the response features are
captured by the model but the temperature dependent response is not predicted;
model constants were chosen for each temperature individually. The model assumes
a linear elastic behavior before the glass yields, while the experiment shows a vis-
coelastic characteristic in the transient range.
The viscoelastic White-Metzner and Marrucci models explored did not capture
the rate dependence response well, although the Marrucci model is able to predict
the stress overshoot. These were examined here because of their similarity to the
Je�rey model used in �ber drawing. Most glass �ber drawing processes occurs in
the high temperature range (above glass softening point), the glass shows Newtonian
behavior in this temperature range. Large deformation during �ber drawing occurs
112
in the high temperature range, thus viscoelasticity has little e�ect on the overall
kinematic and dynamic drawing variables. Although the Je�rey model captures the
�ber drawing process well, it is insu�cient to describe the viscoelastic behavior at
longer times.
BIBLIOGRAPHY
113
114
BIBLIOGRAPHY
[1] F. T. Geyling and G. M. Homsy. Glass Tech., 21(2):95, 1980.
[2] A. K. Varshneya. Fundamentals of Inorganic Glasses. Academic Press, Inc.,1994.
[3] R. H. Doremus. Glass Science. John Wiley and Sons, 1973.
[4] J.H. Simmons; R.K. Mohr and C.J. Montrose. J. Appl. Phys., 53:4057, 1982.
[5] P. Manns and R. Br�uckner. Glastechn. Ber, 61:46, 1988.
[6] K. L. Loewenstein. The manufacturing Technology of Continuous Glass Fibers.Elsevier, 1993.
[7] A. A. Gri�th. Phil. Trans. Roy. Soc. A, 221:163, 1920.
[8] A. Smekel. J. Soc. Glass Tech., 20:4322{453T, 1936.
[9] O. Reinkober. Phys. Z., 32:243, 1931.
[10] J. B. Murgtroyd. J. Soc. Glass Tech., 28:368{405T, 1944.
[11] T. A. Bartenev. Zh. Fizi. Khim., 29 (3):508, 1955.
[12] A. F. Prebus and J. V. Michener. Industr. Eng. Chem., 46:146, 1954.
[13] W. H. Otto and F. W. Preston. J. Soc. Glass Tech., 34 (15):63, 1950.
[14] W. H. Otto. J. Am. Ceram. Soc., 38 (3):38, 1955.
[15] W. F. Thomas. Glass Tech., 1(1):4, 1960.
[16] W. F. Thomas. Glass Tech., 12(2):42, 1971.
[17] K. L. Loewenstein and J. Dowd. Glass Tech.Glass Tech., 9(6):164, 1968.
[18] V. G. Pahler and R. Br�uckner. Glastech. Ber., 58(2):33, 1985.
[19] H. Stockhorst and R. Br�uckner. J. Non-Cryst. Solids, 49:471, 1982.
[20] H. Stockhorst and R. Br�uckner. J. Non-Cryst. Solids, 86:105, 1986.
115
[21] E. H. Andrews. Brit. J. Appl. Phys., 10:39, 1959.
[22] S. Kase and T. Matsuo. J. Poly. Sci. Part A, 3:2541, 1964.
[23] D. A. McGraw. J. Basic Eng., 90:343, 1968.
[24] M. Matovich and J. R. A. Pearson. Ind. Eng. Chem. Fund., 8:512, 1969.
[25] W. W. Schultz and S. H. Davis. J. Rheol., 40:285, 1982.
[26] B. P. Huynh and R. I. Tanner. Rheology Acta, 22:482, 1983.
[27] Y. T. Shah and J. R. A. Pearson. Ind Eng. Chem. Fund., 11:145, 1972.
[28] Y. T. Shah and J. R. A. Pearson. Ind Eng. Chem. Fund., 11:150, 1972.
[29] J. R. A. Pearson and Y. T. Shah. Ind Eng. Chem. Fund., 13:134, 1974.
[30] U. C. Paek and R.B. Runk. J. Appl. Phys, 49(8):4417, 1978.
[31] M. M. Denn; C. J. Petrie and P. Avenas. J. AIChE., 21:791, 1975.
[32] R. K. Gupta; J. Puszynski and T. Sridhar. J. Non-Newtonian Fluid Mech.,21:99, 1986.
[33] T. Sridhar; R.K. Gupta; D. V. Boger and R. Binnington. J. Non-NewtonianFluid Mech., 21:115, 1986.
[34] N. Phan-Thien. J. Non-Newtonian Fluid Mech., 25:129, 1987.
[35] W. W. Schultz. J. Rheol., 31:733, 1987.
[36] D. A. Nguyen; R.K. Gupta and T. Sridhar. J. Non-Newtonian Fluid Mech.,35:207, 1990.
[37] G. K. Gupta; W. W. Schultz; E. M. Arruda and X. Lu. Rheol. Acta, 35(6):584,1996.
[38] X. Lu; E. M. Arruda and W. W. Schultz. J Non-Newtonian Fluid Mech., toappear, 1998.
[39] C. J. Petrie and M. M. Denn. J. AIChE., 22 (2):209, 1976.
[40] W. Weibull. J. App. Mech., 18:293, 1951.
[41] Horst Scholze. Glass. Springer-Verlag, 1991.
[42] M. Matsui and D. C. Bogue. Polymer. Eng. Sci., 16:735, 1976.
[43] G. W. Scherer. Relaxation in Glass and Composites. John Wiley and Sons,1986.
116
[44] R. J. O'Donnell and J. A. Zakarian. Ind Eng Chem Process Des Dev, 23:491,1984.
[45] G. K. Gupta and W. W. Schultz. Int J Nonlinear Mech 33: (1) 151-163 JAN1998, 33:151, 1998.
[46] A. Q. Tool. J. Am. Ceram. Soc., 29:240, 1946.
[47] H. N. Ritland. J. Am. Ceram. Soc., 39:403, 1956.
[48] O.S. Narayanaswamy. J. Am. Ceram. Soc., 54(10):491, 1971.
[49] G. W. Scherer. J. Am. Ceram. Soc., 67:504, 1984.
[50] I. M. Hodge. J. Non-Cryst. Solids, 131-133:435, 1991.
[51] I. M. Hodge. J. Res. Natl. Inst. Stand. Technol., 102:195, 1997.
[52] J. Huang and P. K. Gupta. J. Non-Cryst. Solids, 151:175, 1992.
[53] H. S. Chen and C. R. Kurkjian. J. Am. Ceram. Soc., 66(9):613, 1983.
[54] H. S. Chen. J. Non-Cryst. Solids, 46:289, 1981.
[55] A. R. Berens and I. M. Hodge. Macromolecules, 15:756, 1982.
[56] D. H. Uhlmann and N. J. Kreidl, editors. Glass: Science and Technology. Aca-demic Press, 1986.
[57] J.H. Li and D.R. Uhlman. J. Non-Cryst. Solids, 22:127, 1970.
[58] J.H. Simmons; R. Ochoa; K.D. Simmons and J.J. Mills. J. Non-Cryst. Solids,105:313, 1988.
[59] R. Br�uckner and Y. Yue. J. Non-Cryst. Solids, 175:118, 1994.
[60] Y. Yue and R. Br�uckner. J. Non-Cryst. Solids, 180:66, 1994.
[61] S. Rekhson. J. Non-Cryst. Solids, 131-133:467, 1991.
[62] A. S. Argon. Glass: Science and Technology Vol. 5. Academic Press, Inc., 1980.
[63] A. S. Argon and H. Y. Kuo. Materials Science and Eng, 39:101, 1979.
[64] D. Deng; A. S. Argon and S. Yip. Phil. Trans. R. Soc. Lond., A 329:613, 1989.
[65] M. C. Boyce and E. M. Arruda. Polym. Eng. Sci., 30:1288, 1990.
[66] O. A. Hasan and M. C. Boyce. Polym. Eng. Sci, Feb.:�nd, 1995.
[67] P.H. Mott; A. S. Argon and U.W. Suter. Philos. Mag., 67:931, 1993a.
117
[68] H. Burlet L. Du�rene; R. Gy and R. Piques. J. Rheol., 41(5):1021, 1997.
[69] D. Acierno; F. P. La Mantia; G. Marrucci and G. Titomanlio. J. Non-NewtonianFluid Mech., 1:125, 1976.
[70] U.F. Kocks; A.S. Argon and M.F. Ashby. Thermodynamics and kinetics of slip,progress in materials science. 1975.
[71] A.S. Argon. Phil. Mag., 28:839, 1973.
[72] M.C. Boyce; D.M. Parks and A.S. Argon. Mech. of Materl., 7:15, 1988.
[73] D. A. McGraw. J. of the Amer. Ceram. Soc., 35(1):22, 1952.
[74] J. L. White and A. B. Metzner. J. Appl. Polym. Sci., 8:1367, 1963.
ABSTRACT
EXPERIMENTAL AND ANALYTICAL INVESTIGATION OF
NON-ISOTHERMAL VISCOELASTIC GLASS FIBER DRAWING
by
Xiaoyong Lu
Chairperson: Ellen M. Arruda, Co-Chairperson: William W. Schultz
A preform glass �ber drawing apparatus is designed and built to investigate non-
isothermal viscoelastic glass �ber drawing. The e�ect of the drawing parameters on
glass �ber properties is studied experimentally and analytically.
Birefringence is a measure of anisotropy in glass structure that can in uence
�ber properties and performance. Birefringence is produced during �ber drawing
as the �ber is rapidly stretched in the viscoelastic glass transition range, and is
\frozen" into the glass during rapid cooling. With our drawing apparatus using
Borosilicate glass (Corning code 7740) preforms, we produce glass �bers for a range
of process conditions and measure their as-drawn birefringence. The development of
birefringence in glass �bers is found to depend on the amount of deformation, the
deformation rate, and temperature. Results for various process parameters show that
increasing draw ratio, increasing elongation rate, and decreasing draw temperature
increase birefringence.
Post-process annealing is used to examine the time and temperature dependence
of glass �ber birefringence relaxation and its e�ect on glass �ber tensile strength.
Birefringence is found to completely relax in the temperature range close to the glass
transition range as expected, but it is also noted that birefringence shows substantial
(although incomplete) relaxation in a temperature range well below the glass tran-
sition. This low temperature relaxation indicates that the relaxation process is due
to a very wide distribution of relaxation times.
A nonisothermal one-dimensional model is examined to simulate the glass �ber
drawing and �ber birefringence relaxation using a generalized Je�rey model. The
relaxation time and retardation time are both functions of the temperature. The
conservation equations of mass, momentum and energy are solved simultaneously
with the viscoelastic constitutive equation to determine the spatial distribution of
�ber dimensions, velocities, stresses and frozen in elastic strains. The e�ects of
process parameters on the �ber properties are simulated. The model predictions are
compared with the experiments and are found to be in qualitative agreement with
data.