INVESTIGATIONS
AND
OF
ANELASTICITY
THE
STRENGTH
OF
GIASS
by
ALI SUPHI ARGON
B.S.,
Purdue University
(1952)
S.M.,
Massachusetts Institute of Technology
(1953)
SUBMITTED
OF
THE
IN
PARTIAL
FULFILLMENT
REQUIREMENTS
DEGREE
OF
FOR
DOCTOR
THE
OF
SCIENCE
at the
MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Signature of Author.. .
..
w....W*V.
....................
Department of Mechanical EJineering, May 14, 1956
Certified
by...................................................
Thesis Supervisor
Accepted by ....
............
Cha rman, Departmental Co
'1/
ttee on Graduate Studentd
"Investigations of the Strength and Anelasticity of Glass"
by Ali Suphi Argon
Submitted to the Department of Mechanical Engineering on May 14, 1956
in partial fulfillment of the requirements for the degree of Doctor
of Science.
ABSTRACT
By use of the Hertz ball indentation method, the strength of glass
is found to be form one to nearly two orders of magnitude higher than
those observable in large specimens by conventional testing methods.
This is due to the very small areas subjected to high stress. The method
is very useful in evaluating the quality of glass surfaces, and comparing
different surface treatments. It can also give some qualitative information about the distibution of Griffith cracks on the surface.
The anelasticity of a pyrex glass is studied by its elastic creep
at various temperatures. The low temperature viscosities are evaluated
and are found to be of the order of
poises around 2000 C.
By a step by step thawing of the recoverable delayed elastic strain put
into the specimen by prior creep at an elevated temperature, the activation
energy spectrum for the deformation process is obtained. The peek of the
spectrum is around 48,000 calories per mole.
1019
A differential refractometer for the direct measurement of residual
stress on tempered glass is described, and its potentialities in the
field of photoelastic stress analysis are indicated.
Thesis Supervisor: Dr. Egon Orowan
Title: George Westinghouse Professor of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
May 14, 1956
Professor J. Keenan
Chairman, Department Committee on the Graduate School
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
Dear Sir:
In partial fulfillment of the requirements for the degree of
Doctor of Sceience, I hereby submit a thesis entitled "Investigations
of the Strength and Anelesticity of Glass".
Sincerely,
Ali Suphi Argon
TABLE
OF
CONTENTS
PART
I
CHAPTER
I.
II.
III.
IV.
PAGE
INTRODUCTION.
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General Testing Procedure
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Chemically Polished Glasses
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60
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PREVIOUS RELATED STUDIES . .
DESCRIPTION OF APPARATUS
.
EXPERIMENTAL RESULTS
.
.
Drawn Sheet Glass
Plate Glasses
. . . . . .
Special Glasses . .
Position of Fracture
V.
DISCUSSION OF RESULTS
.
.
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0..
The Size Effect of the Fract ure
Position of Fracture
VI.
CONCLUSIONS
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Strength
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7
iii
PART
II
CHAPTER
VII.
PAGE
INTRODUCTION .
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The Problem.
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Definition of Terms Used
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PREVIOUS REIATED STUDIES
Early Studies
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Statement of Organization.
VIII.
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68
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69
71
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Description of the Liquid State of Glass
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71
73
Recent Theoretical and Experimental Studies of
the Elastic After Effect and Internal
. . . . . . . . . . . . . . . . .
Damping
IX.
THEORETICAL TREATNENT OF THE ELASTIC AFTER EFFECT
Time Laws
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76
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79
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79
Determination of the Activation Spectrum from the
...........
Creep Curves Directly
Recovery Creep
X.
.
0
DESCRIPTION OF THE APPARATUS
The Creep Apparatus
..
.* .
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88
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91
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92
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92
CHAPTER
PAGE
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92
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92
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97
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97
Method of Preparation of Specimens . . .
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97
Inspection and Measurement of Specimens
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99
Requirements
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Design Information .
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Calibration of the Torsion Spring
The Specimens
XI.
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EXPERIMENTAL RESULTS
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104
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106
Determination of the Activation Energy Spectrum .
111
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General Description of Tests
Effect of Temperature on the Deformation
XII.
DISCUSSION OF RESULTS
. . . . .....
. . .0
The Temperature Dependence Tests
Activation Energy Spectrum
XII I.
CONCLUSIONS
.
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PART
XIV.
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116
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116
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117
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121
III
DIFFERENTIAL REFRACTOMETER (STRESS METER)
. . * .
1
2
124
CHAPTER
PAGE
The Problem
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123
Statement of the Problem
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124
Definition of terms Used
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124
Theory
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124
Importance of the Problem
Fundamental Theory . . .
XV.
.
NETHOD OF PROCEDURE
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127
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Critical Angle Refractometers
127
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Differential Refractometer .
XVI.
DESIGN DETAILS OF THE STRESS NETER
Refracting Prism
The Telescope
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. . .
Procedure of Calibration
XVIII.
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136
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136
138
CALIBRATION OF THE STRESS METER .
Accuracy
130
. .......
Other Optical Parts
XVII.
123
.
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CONCLUSI01S . . . . . . . . . . . .
BIBLIOGRAPHY . . . . . . . . . . L . . . . . .
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138
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139
142
144
146
vi
PAGE
CHAPTER
APPENDIX
I
APPENDIX
II
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.a
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150
152
OF FIGURES
TABLE
FIGURE
PAGE
.
1.
Monotron Hardness Tester
2.
Cross section of the optical table
3.
-
20.
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...
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8
.......
.
..
Fracture Histograms of Various Glasses . . .
..
..
. . .
.
9
.
13-16
.
18-31
21.
Change in the Mean Strength across the Path of a Polishing
wheel
22
43.
-
42.
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....
........
.
33
......
.
Fracture Histograms of Chemically Polished Glasses . . . 34-54
Frequency of Fracture at a Distance Away from the Edge of the
Contact Circle . . . . . . . . . ............
55
44.
Increase in Strength with Decreasing Indenter Diameter
45.
Theoretical Likelihood of Fracture as a function of position
46.
The Three Components of Deformation
47.
Radial Distribution in Vitreous Silica
48.
Abrupt Change in Property due to Rapid increase in Relaxation
Time
49.
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59
61
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70
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75
Free Energy Diagrams for an Ion in a Virgin Structure and a
Stressed Structure
76
.. .. .. . . .. . . .....
.
50.
Schematic View of the Creep Device
51.
Recording System
52.
Side View of the Creep Apparatus
53.
Filament Drawing Device in Compressed Position
54.
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96
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.*..........
93
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98
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100
Filament Drawing Device in Extended Condition . . . . . . . .
100
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viii
PAGE
FIGURE
101
55.
Tubu2ar Torsion Creep Specimens . . . . . .
.
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..
56.
Photomicrograph of Typical Specimen . . .
.
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*
57.
Contribution to the Angle of Twist of the End of the Specimen
103
58.
Delayed Elastic Creep at Various Temperatures
107
59.
Variation of Viscosity with Temperature . . . . . .
.
.
.
108
.
Step by Step Recovery of a Pre Strained Specimen
61.
Activation Energy Spectrum of Pyrex Glass . . . . . .
62.
The "Eating Function"
63.
A partially Used Up Spectrum of Activation Energies .
64.
Refracting Prism of a Refractometer .
.
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.0
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*.
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112
60.
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101
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119
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0
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119
128
. . .
.0
65.
The Abbe Refract~meter
.
129
66.
A Cross Section and a Side View on the Stress Meter
131
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132
67.
Stress bands in the Field of View of the Stress Meter
135
68.
Refracting Prism of the Differential Refractometer
128
69.
Calibration Set Up of Stress Meter
.
140
70.
Calibration Curve of the Stress Meter . . . . . .
141
71.
Indentation Histogram
72.
Variation of the Radial Tensile Stress at Various Depths
73.
Locus of Zero Radial Stress .
0
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151
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153
154
GENERAL
ORGANIZATION
OF THE
THESIS
This thesis comprises three parts. The main body consists of two
sets of investigations treating the subjects of surface strength, and
anelasticity and viscous properties of glasses. The third part of the
thesis is on the design of a differential refractometer (stress-meter)
for the determination of the residual surface stress on tempered
glass. These three subjects will be treated in the order introduced
above.
i
PART
SURFACE STRENGTH
I
OF
GIASS
CHAPTER I
INTRODUCTION
The low strength of glass is generally ascribed to the presence of
Griffith cracks. The experimental determination of the strength of
glass thus entails a large number of observations to get a statistically
valid result. In most of such experiments and in the majority of cases
in practice, however, fracture is caused not by Griffith cracks on the
surface but from the more severe edge cracks resulting from the diamond
scratch which is necessary for cutting. To determine the true strength
of glass one must therefore, carry out tests in such a way that the
cut edges are not under stress. There are a number of different methods
of accomplishing this. The one most suitable for plate glass, however,
is the ball indentation method. In view of the fact that it is also
possible to test very small areas by this method by using increasingly
smaller balls, a very qualitative picture of the distribution of the
Griffith cracks can be obtained. The method is also useful in evaluation
of various surface treatments.
CHAPTER
I. PREVIOUS
II
REIATED STUDIES
The elastic problem of two spheres in contact was solved by Hertz
Auerbach
was the first one who experimentally verified Hertz's
solution, and attempted to use the indentation method as a tool for
determining material properties. Among other things Auerbach observed
that the fracture stress was quite variable with one particular indenter,
and the fracture stress increased in general for indenters with small
radii, (Auerbach carried out his experiments by impressing lenses of
known radii onto plate glass). He also observed that cracks never formed
at the radius of contact where the Hertz solution predicts the highest
surface stress, but on the average they formed on circles with 19% larger
radii.
More recently Powell and Preston
experimented on a large number
of different glasses with various surface treatments, and found that
the fracture stresses for some of their specimens were only at the most
one order of magnitude smaller than the estimated theoretical strength.
The above investigators also observed an increase in strength after etching
(1)Hertz, H., Gesammelte Werke, vol. 1, p. 155
(2)
Auerbach, F., An
Ph. Chem., vol. 53, p. 1000
(3)Powell, H. E., F. W. Preston, J. Am. Cer. Soc., vol. 28, pp. 145-49
5
the surface with dilute hdro-fluoric acid.
Several other investigators observed slight increases in strength
when the test was carried our under oil, and decreases of strength when
the tests were carried out on a wet surface.
II.
DESCRIPTION
OF
THE STRESS
FIELD
When a sphere and semi infinite solid of the same material are in
contact and pressed together with a load P, the stresses in the semi
infinite solid are given in cylindrical coordinates as follows:-,4
-
+ *[zg.
?
Jr-%t1j-
(2)
+F t[I4(
(7Jjy~l4
323
L4%(
T4+(47
4
ik2-Wit
(/+Huber, H., Ann. der Physik, vol. 14, p. 153
(3)
(
(4)
where,
pt = 3P
/
27a
2
a
= radius at the contact circle
P
= applied compressive load
u =(r
2
+ z2
a2
r + z
a2
+ 4 a2
)
Of these stresses the only one of importance is the radial tensile
stress. Since fracture starts from the free surface, the radial stress
on the surface would be the logical parameter to be correlated with
each fracture
Thus for z = 0 the significant radial stress is simplified
z2r
r-
(5
(5)
It is to be noted here that P now refers to the magnitude of the
compressive load.
In reality fracture results from a Griffith crack the root of which
lies below the surface.
CHAPTER
DESCRIPTION
III
OF APPARATUS
The experiments were performed on a converted Shore Monotron Hardness
tester which can be seen in Fig. 1. In place of the standard lever arm
of the above tester, a worm and wheel drive was installed for more
uniform loading. Furthermore the loading table was replaced by an "optical
table", permitting observation of the area under load at all times
through a microscope. When fracture took place the diameter of the crack
circle could be read directly by means of a pre-calibrated reticule
inserted at the back focal plane.
The tests were carried out in the following manner: the head of the
instrument containing the load indicator was lowered until contact was
established between the indenter and the glass plate. Illumination at
a grazing incidence to the top of the glass showed the area of contact
as a bright spot in a dark field. As loading commenced the bright area of
contact grew until suddenly a circumferential fracture appeared in the
field enveloping the contact circle. The crecks start concentrically
around the area of contact, and fan into the glass, away from the indenter.
After penetrating the plate for a certain distance the cracks stop,
tracing out a truncated cone shaped piece of glass, joined to the parent
plate along its base. The magnification of the microscope was 80 times,
and proved to be sufficient for all the indenters. Fig. 2 shows a cross
41A
%#A.
Fig.
1
MonotrQn Hardness Tester
Gloss
Microscope
indenter
Optical table
Fig. 2 Cross section of the optical table%1
=ARM==
10
section of the "optical table".
CHAPTER
EXPERIMENTAL
IV
I. GENERAL
TESTING
RESULTS
PROCEDURE
The specimens were flat pieces of glass cut to 4 x 4 or 6 x 6 inch
size. Their thickness varried from 7/32"1 for sheet glass to 1/2"1 for plate
glass, the majority being around 1/4" and 3/8" thick. For thinner pieces
of glass bending stresses may become appreciable which may not only cause
erroniously high Hertz stresses, but in extreme cases also fracture,
starting from the other side.
The majority of the different glasses were supplied by Pittsburgh
Plate Glass Co. Every possible precaution was taken by the suppliers in
packing, shipment and storage og glasses to preserve the surface in the
best possible condition. Aside from this protection from mechanical
injury, however, so special pains were taken to eliminate natural atmospheric
attack.
Plates were indented only on one side from fear that the other side
may have been damaged from contact and pressure from repeated indentations.
Whenever it was feasible, one hundred indentations were made on each
plate. Occasionally the size othe plate or the peculiarity of the particular
test did not permit this many indentations.
641-
56k
481
40 -
32-
24
161-
I
0 0
0
rf)
I
0
0
o
'
FRACTURE
F~.3
a
0
*C
STRESS,
SZ)/sfrisofior
04 "',.
/E16 fve
00
0m
0
Kg/
mma
Sheet
~/E~S
kI'7q~
641-
561-
48 V
uJ
0
z
w
0
40
32-
LA-
0
z
u
u-
2416
8-
r-"FhA
-N
o0
tp)
0
'T
~
0
in
FRACTURE
pI~.
Cl4 SS
0
%D
oo0
tl
CD
STRESS,
0
(n
0
Kg/ mm1
&'/p~nfA Ztr~donfrue
y I
0
-1
0
0
o
N ( iJ O
641
561-
481-
401-
32F
24-
o
_
o
K)
9
o
-
f
0 0
t
0
M
FRACTURE
STRESS,
S-Irt 05 h Dl
ot;o n
iide#ikr,
Svtr/we
eleheol
rY1A111
0
O
0
Kg/ mmz
WY
/0r1r
0?t
64
FRACTURE
1+
14d-o
Kg/mmt
STRESS
i A I,//&
Z/*ll' /
ft
SusyIace efchAed/
w
01
A%/
Atflo.
/;l
/ !Pop11F.
At
16
The results are presented as frequency plots of the fracture stresses.
Althpugh this method of presentation is the most convenient it does not
tell anything about the position of fracture in relation to the contact
circle.(*
Several plots of considerably greater complexity were tried to
incorporate this parameter into the results. Since there is no simple way
of evaluating the distance of fracture data to furnish an idea about the
crack distribution function, the greater pains required for preparation
of these plots was considered not to be worthwhile for only the most
general gain of qualitative information. Nevertheless, one of these plots
is included in Appendix I as an item of interest.
The indenters which were used in these tests were 1/16", 1/8", and
3/16" diameter ball bearing balls.
II.
DRAWN
SHEET
GIASS
Indentations were performed on 7/32" sheet glass with two indenters
only, due to the fact that the large ball often required a load in excess
of the capacity of the load gauge of the Monotron. The results are shown
in Figs. 3 and 4. A set of experiments in which the surface was progressively
etched away with dilute hydro-fluoric acid was also performed, (see Figs.
5 and 6).
The frequency plots show that on the average sheet glass is very
(*)The position of fracture is important in the sense that it conveys
some qualitative idea about the characteristics of the crack distribution
function.
17
strong, with a good part of the distribution curve lying above the 100 Kg/mm 2
mark.
A significant drop in the mean strength for the 1/8"D indenter can
be seen in Fig. 4. This could mean that the larger cracks are scarce, and
that the distance between them is of the order of the contact circle of
the larger ball.
A more striking result, however, is the drastic shift of the mean
strength and the small standard deviation for the hydro-fluoric acid etched
sheets. A comparison of the etched and unetched sheets shows that the acid
has attacked preferentially the larger cracks and has inactivated them
by changing their geometry in such a way as to reduce the high stress
concentration factor.
III.
PLATE
GLASSES
The results with the 3/8" and 1/211 plate glasses indicate that they
are considerably weaker than sheet glass. The rise in strength with smaller
indenters is also not as marked as in sheet glass meaning that the crack
distribution is more uniform and the concentration is denser. No etching
experiments were carried out with these glasses. Figs. 7-12 present the
results of these tests.
IV. SPECIAL
GLASSES
A phosphate glass, a plate of fused silica, and some #7740 pyrex glass
64
56
48
40
32
24
16
S
x
0
re)
0
V
o
0
FRACTURE
-7
/ig .Z7 /.'7de Ser
4
Z)
0
0
co
00
ao
STRESS,
2)sisr;6ui1,o,
o
0
4
0
rn
0
'2
0
ut)
Kg/ mm2
0/
//
C;/0 ss w r/hP
64
56
48
o40
z
0
LJ
32
U-
0
24
a~
16
8
0
0
-
0 00
1, r4
0
00
FRACTURE
& ~ ~ ~
~
NHk /2"
0
to
u
000
-
STRESS,
/r
0 0
ro
00
9
Kg/ m m
~~~?Z .l.Weg4Drlano
>. /a'e
N
"Pt ass
64
561
481-
401
32 1
24[-
16
K
-
cuI
o
IT
o
U')
o
to
o
00
C-
STRESS,
FRACTURE
F&9'
0
3-/reillus fri6~,A,
w/A,
J/
4
P
o
Kg/ mm
3A
4b%"r
P/4e ;/0ss
64
5u-
48
40
32
24-
Ka
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h)
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reK
vin
was tested in this group. The common characteristic of these glasses was
their uniformly low mean strength. No fracture strength more than 80 Kg/mm2
was observed with any of these of these glasses. Figs. 13-17 give the
results of these tests.
A set of etching experiments with pyrex, however, gave some interesting
results. As a consequence of progressively longer etching, the narrow
distribution curve with its mean value at a low 26.9 Kg/mm2 began to
spread and show a shift of the mean strength toward higher values. This
indicates that the initial distribution of these large cracks probably
retained its shape while the progressive ething reduced their number.
A shift of the mean strength upward and an increase in the standard
deviation was theoretically deduced by Fisher and Hollomon (5)Figs. 18-20
give the results of etching.
V.
CHEICALLY
POLISHED
GLASSES
A set of chemically polished glasses comprising a strip cut transversely to the forward motion of the polishing wheel were kindly furnished
by the Pittsburgh Plate Glass Co. Visual inspection of the side plates
showed insufficient polishing, whereas the plates coming directly from
under the center of the polishing wheel showed a high degree of polish.
The surface strength results followed in general the visual observations,
The mean strength was low at the edge plates. It increased toward the
(5)Fisher, J. C., J. H. Hollomon, Metals Technology, vol. 14, T.P. 2218
100
E
E
180
C)
60
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Fig. 21
4
5
6
7
NUMBER
Change in mean strength across path of polishing
wheel
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r/
43
Fig.-- Frequency of fracture at a distance away from
the edge of the contact circle
2.0
56
interior and reached its maximum for the central plate. A regular rise
in the fracture strength with decreasing ball diameter was observed
which was similar to that in plate glasses. Fig. 21 shows the ehange in
mean strength across the path of the polishing wheel.
In general the fracture strength of these chemically polished glasses
compared favorably with sheet glass, and were definitely superior to
mechanically polished plate glasses. Figs. 22-42 show the results in
detail.
VI. POSITION
OF FRACTURE
A test was carried out to determine the frequency of the position of
fracture in relation to the contact circle. For this experiment second
rate plate glass was used which had been stored in the laboratory for
a long time with no precautions. The test consisted of recording the
diameter of the contact circle as well as the fracture circle. The
frequency of fracture as a function of r/a is illustrated in Fig. 43.
CHAPTER
DISCUSSION
I. THE
SIZE
EFFECT
V
OF
OF
RESULTS
THE
FRACTURE
STRENGTH
The general increase in fracture strength for indenters with
small diameters can be explained partly by the fact that the area under
high stress is smaller for smaller indenters. Thus, it is more probable
to find a dangerous crack in a larger area which will give a low fracture
stress.
If a size effect relation for the change of strength with area
K/A (
=
(6)
is assumed to hold, where
a base strength, engineering strength
K = a constant
A = the area under stress,
then, the following analysis relating the fracture stress to ball diameter
can be deduced.
The stressed area is
A =7r(Ca)
2 _
7ra
2
22
C-1)7r a2
This would be the kind of expression which was experimentally observed
to hold for thin glass fibers by Karmarsh.
where, a = the contact circle radius
C = a constant,
if the stressed area is considered arbitrarily bounded at a certain circle
with radius Ca. For a given load the relation
( a/r )2
follows from the Hertz formula,(5). Since, however,
and
A = C7a 2 ,P
a = K1 D1/3
it follows that,
-%=K 2 / D2/3
T
where,
(7)
r = distance from the center
C1 = a constant
K = a constqnt
K2 = a constant
D = diameter of the indenter
From the above equation, the effect of different indenters can be formulated
as follows:
(TI - Tg)/( 9~i -
-) = (
2/D1
)2/3
(8)
If now C is neglected in comparison with the Hertz stresses,
T/(Qj= (
D2 /D1 )2/3
(8a)
In Fig. 44 the D /D3 ratios are plotted against a-/(T3 for several
2.2
2.0
1.8
1.6
U.
0
1.4
0
I4
cc)
1.2
1.0
0.2
0.4
Di/D 3 , RATIO OF
Fig. 44
0.6
INDENTER
0.8
1.0
DIAMETERS
Increase in strength with decreasing indenter diameter
60
glasses together with relation (8a). The index 3 here stands for the
3/16"D ball. Although the agreement is poor, there is a general trend
roughly following the theoretical curve.
II. POSITION OF FRACTURE
A qualitative explanation for the appearent paradox that fracture
never takes place where the surface stress is highest can be given upon
inspection of the stress field and the use of the Griffith criterion for
fracture.
As in the case of a point load applied perpendicularly to the
surface of the solid, in the Hertz experiment there is a spherical volume
under the area of contact within which all normal stresses are compressive.
Thus, it becomes clear that all potential cracks outside the circle of
contact can not grow, even though the surface stress may be sufficiently
large, because the root of the crack could be embedded in the compressive
region.
The Griffith criterion states that if the rate of release of elastic
energy is equal to the rate of increase of surface energy with respect
to the length of the crack, the crack may grow in equilibrium. Since the
rate of release of elastic energy is linear in a uniform stress field,
whereas the rate of increase of surface energy is a constant with respect
to the crack length, beyond a cextain length of crack the situation becomes
unstable, and the crack would accelerate. Let this difference between the
respective rates be denoted by a finction V
. Although in a uniform
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
r,/
Fig. 45 Theoretical likelihood of fracture as a function of
position
0
stress field
q
will be the same at all points of the material for a
crack of a certain length, in a complex field like that in the Hertz
case the function
will vary from point to point. The Griffith criterion
would then predict that given a chance, a crack would prefer to run in a
field where the function O is maximum. A consideration of this sort for
a particular case gives the curve seen in Fig. 45. This curve was obtained
by assuming that the final outcome would be influenced by the function
at a depth of one micron which corresponds to the order of magnitude for
the length of the average Griffith crack for this particular glass. If
the glass had a greatly different crack distribution function, the function
may have to be evaluated at a different depth which would give a
shift of the peak. For the details of the calculation of this curve the
reader is refered to Appendix II. A comparison between this theoretical
curve and the experimental distribution plot shows good agreement.
CHAPTER VI
CONCLUSI01E
The ease with which results were obtained made it clear that the
Hertz experiment is ideally suited for the determination of the surface
strength.
The results 6f tests can be summarized briefly as follows:
1.
Rapidly cooled sheet glass which has had the minimum of mechanical
handling shows the highest strengths.
2.
Mechanically polished plate glass is somewhat weaker than
sheet glass and seems to have a uniform distribution of cracks
which must have been introduced as a result of the polishing
treatment.
3.
Chemically polished plate glasses showed large scatter in strength,
had occasional high strengths comparable with the highest in
sheet glass, and were in general stronger that mechanically
polished plate glass.
4.
Glasses with special compositions and fused silica had very low
strengths. The high resistance to thermal shock of pyrex and
fused silica are not because of the abscence of dangerous cracks
but because of their low coefficient of thermal expansion.
5.
Etching in 1% dilute hydro-fluoric acid eliminated the few
dangerous cracks in sheet glass, giving a narrow resultant
distribution curve with a shift in the mean strength toward a
higher value. Ething of pyrex produced a general spread of the
strength distribution as well as a shift of the mean strength
upward. The results of etching may explained by a preferential
elimination of the larger cracks.
6.
Due to the larger area under stress, a lower strength was
observed for indenters with large diameters.
7.
Fracture is never observed at the contact circle where the surface
stress is the highest, but always at a somewhat larger diameter.
An explanation for this behavior can be given by considering
the stress field in depth and applying the Griffith criterion
for fracture.
8.
In general the order of magnitude of the strength of glass
observed in these experiments is one order higher than the
engineering strength, and in the best cases only one order
smaller than the high values observed by Griffith on virgin
glass.
PART
ANEIASTICITY
GIASS
II
AND
AT
LOW
VISCOSITY
OF
TEMPERATURES
PYREX
CHAPTER VII
INTRODUCTION
It is now common knowledge that a perfectly elastic material in
which strains corresponding to stresses are attained in the interval of
time that is required for an elastic wave to travel through the material
does not exist. Apart from the irreversible processes of plastic deformation and viscous flow, the recoverable elastic deformations which are
variously called elastic-after-effect, elastic creep, delayed elasticity,
or anelasticity have for a long time attracted the attentions of many
physicists and engineers as a tool for the study of the structure of
matter.
I.
Object:
THE
PROBLEM4
The object of this investigation was to study the non-elastic
deformations in glass as a tool for a better understanding of the
structure of glass. Since the elastic-after-effect and viscosity are
strongly influenced by the state of the structure, an investigation of
the variation of these properties as a funstion of stress, temperature,
and degree of stabilization would be of interest.
Importance of the study: Glass is a material which has reached its
low temperature solid state in a continuous way froii the melt without
undergoing any abrupt phase change characteristic of pure substances.
-----
MEOW
67
The appearent solidity is achieved by a rapid increase of the viscosity
with droping temperature. Thus, below the transformation region the
relaxation times accompanying the viscosity are several orders of
magnitude larger than the time spent in most experiments. Therefore,
glass is considered as a rigid material although in reality it is a subcooled liquid with an enormously high viscosity. Previous investigations
of the viscosity aid the elastic-after-effect by a great many observers
have established that glass is by no means a simple liquid, but instead
has the character of a giant molecule. In view of this complexity of
behavior, few investigators have gone beyond a simple phenomenological
observation. This study was undertaken with the intent to bridge this
gap between experimental observations on the one hand and theoretical
models on the other.
II.
DEFINITION
OF TERMS
USED
The three components of deformation: When glass is subjected to stress
it deforms in manner as illustrated in Fig. 46. This curve can be broken
into three components which more or less stand for three different
mechanisms. The abrupt instantaneous rise corresponds to the elastic
deflection which reaches its ultimate magnitude in an interval of time
required for an elastic wave to travel through the material. On top of the
elastic component there is a transient part which reaches a horizontal
asymptote. This deformation is termed the elastic-after-effect. The third
component is the Newtonian viscous flow. Various mechanical models
composed of springs and dashpots can give creep curves exhibiting these
±
+
TIME
Fig. 46
The three components of deformation
69
three components of the deformation. If their purely arbitrary nature
is not forgotten these models could be very instructive in obnstructing
creep curves having desired shapes. When this process is carried too
far, however, there is a danger of visualizing seperate phases with
differing elastic moduli and viscosities in an othervise homogeneous
and isotropic material. Therefore, such mechanical models will not be
utilized here; instead processes will be explained by kinetics of
structural configurations which according to the present state of knowledge
have a high probability of representing the true state of the matter.
The reader who is interested in theoretical treatments of the delayed
elastic and viscous deformations by means of mechanical models will find
no shortage of excellent accounts in the literature.
III.
STATEMENT
OF
ORGANIZATION
The remainder of this study is treated as follows:
Chapter VIII presents a survey of previous related studies of the
subject.
Chapter IX is devoted to theoretical treatments of the elastic-aftereffect.
Chapter X describes the apparatus which was used to carry out the
experiments, and for the preperation of the specimens.
Chapter XI consists of a presentation of the experimental results.
Chapter XII is a discussion of the experimental results.
Chapter XIII formulates the conclusions of this study.
CHAPTER VIII
PREVIOUS
RELATED
STUDIES
I. EARLY STUDIES
Perhqps the first account of anelasticity is given in the papers of
Weber(6, 7)who in 1835 carried out experiments on silk filaments and
glass fibers. These experiments were prompted by an imperfect behavior
of materials used in suspensions of electrical measuring devices in which
the pointer of the instrument did not attain its final reading immediately,
but, after an instantaneous deflection gradually crept toward it, and
upon removal of the electrical potential the return to zero reading
exhibited the same phenomenon in reverse. Weber's investigation disclosed
that the delayed deformation beyond the instantaneous elastic one was
wholly reversible, and therefore, was of a different character from
irreversible processes like plastic and viscous deformations. Thus Weber
corrected the notion of imperfectly elastic behavior and coined the
expression "Elastische Nachwirkung" or elastic-after-effect to describe
this phenomenon.
After Weber, the father and son team of F. and R. Kohlrausch devoted
many years of their lives to investigations of such delayed phenomenon
(6)Weber, W., Pogg. Ann. Physik, p. 247, (1835)
(7)Weber, W., Pogg. Ann. Physik, p. 24, (1841)
on raw silk and glass fibers, silver wires and rubber threads.(8
9, 10, 11)
R. Kohlrausch, in study of residual charges in Leyden jars, discovered
a fundamental relationship between delayed elastic effects and dielectric
properties.(12) Following these pioneers, many other investigators, among
them Bolzman, Volterra, and Becker tried to deduce time laws of deformation
that would fit experimental results, and also explain such strange
behaviors as the extraordinary memory effects which were observed by
F. Kohlrausch. The concern of most of these investigators was to determine
the time law of the delayed elastic effect by sophisticated curve fitting.
This resulted in a wide variety of time laws, from exponential through
power laws to logarithmic laws for different materials.
The present more refined viewpoint of atomic configurations with
alternate free energy troughs has its origin in the x-ray investigation
of the structure of glass by Warren and his collaborators.(13)
The small angle scattering experiments of Warren have shown that
statistically the nearest neighbor relation in glasses between network
formers and oxygen is the same as in the crystalline phase. The radial
(8)Kohlrausch, R., Pogg. Ann. Physik, p. 293,
(1847)
(9)Kohlrausch, F., Pogg. Ann. Physik, p. 337. (1863)
(10)Kohlrausch, F., Pogg. Ann. Physik, pp. 1, 207, 399, (1866)
(11)Kohlrausch, F., Pogg. Ann. Physik, p. 337, (1876)
(12)Kohlrausch, R., Pogg. Ann. Physik, pp. 56, 179, (1854)
(13)Warren, B. E. H. Krutter, 0. Morningstar, J. Am.
p. 202, (1936)
er Soc.., vol. 19,
distribution function of vitrous silica as obtained from the experimental
scattering curve is shown in Fig. 47. Taking a silicon ion as the center,
the position of the neighboring ions in the crystalline phase is indicated
by the lines on the top of the figure. The agreement of the first four
peaks isremarkably good.
Evidently then, in a glass on the average the same neighboring
relationships an in the crystalline phase are retained , but no long range
order is established. The x-ray investigations thus verified the homogeneity
of the glass down to the atomic scale, and led to treatments of glass as
an associated liquid.0(14)
II. DESCRIPTION
OF
THE
LIQUID STATE
OF GIASS
(16)
(15)
The modern theories of the liquid state of Bernal,
Frenkel,
et. al. incorporate the concept of configurational order into the older
theories which considered liquids as a collection of randomly oriented
spheres having no directed bonds between them. It was thus recognized that
there is a configurational contribution to the specific heats of liquids.
With an increase in temperature the configuration changes to one of lower
order causing an increase in entropy. Although there is a unique equilibrium
(14)Douglas, R. W., J. Inst. Glass. Tech., vol. 31, pp. 50-89,
(1947)
(15)Bernal, J. D., Trans. Faraday Soc., vol. 33, p. 27, (1937)
(16 )Frenkel, J., Kinetic Theory of Liquids, Oxford University Press, (1946)
73
0
12 -
0
0
i5
0
5
10 -
z
CO
a6.J
4
2 -
of
O
f
R
Fig. 47
4
3
2
in
5
A
Radial distribution in vitreous silica
6
7
74
configurational order for every temperature, the establishment of this
order takes some time. Clearly, the process of changing configurational
order is one of self diffusion which would have a characteristic activation
energy. The relqtion between the mean period
'T
of this diffusion process
and the activation energy A, is given by the relation
=
where
C eA/kT
(17)
(9)
50 is the vibrational period of the diffusing ions. As the temperature
drops the mean period of self diffusion soon becomes pf larger orders than
the intervals of time spent in average laboratory experiments. Evidently,
then below a certain narrow temperature region the time for configurational
equilibrium becomes very large, and the elevated temperature structure
of high disorder is frozen in. This results in a well defined change in
the general trend of physical properties as a function of temperature at
this narrow region. See Fig. 48 (after G. 0. Jones).(18) The narrow temperature region which can be shifted up or down depending upon the duration
of the observation is called the transformation region. In most glasses this
range in around 5500 C. Given sufficient time at a certain temperature
below the transformation region, the property will slowly sink to the
equilibrium value given by the broken line. This process is called stabilization
in glass.
The fact that the configurational order was a unique property of the
(17)Frenkel, J., Op. ct.,
p. 14
(18)Jones, G. 0., Reports od Progress in Physics, vol. 9, p. 136,
(1948)
/
/
TEMPERATURE
Fig. 48 Abrupt change in property due to rapid
increase in relaxation time
76
glass at a given temperature was clearly demonstrated by Lillie's(19)
measurements of the viscosity of glasses below the transformation temperature. In these measurements Lillie showed that the equilibtium viscosity
corresponding to the configurational order is reached from both higher
and lower configurational temperatures. Changes in viscosity and electrical
conductivity toward and equilibrium value were also observed by Peyches. (20)
In practice it has long been known that stress relaxations did not
follow the simple Maxwellian relation which is based on an unchanging
viscosity. Similar changes in the magnitude of the elastic-after-effect
has been observed by many investigators.
III.
STUDIES
RECENT
OF
THE
THEORETICAL
ELASTIC
AND
AFTER
EXPERIMENTAL
EFFECT
AND
INTERNAL
DAMPING
Modern theoretical treatments of the elastic-after-effect are
modifications of theories of viscosity. Since the mechanisms for the two
processes are similar, the considerations are justified.
Perhaps the first one to realize that the elastic-after-effect is
due to a sum of different relaxation systems was Wiechert 21) He assumed a
(19)Lillie, H. R., J. Am.
Cer. Soc.,
vol. 16, p. 619, (1933),
also Ibid.,
vol. 17, p. 43,~(1934), also Ibid., vol. 19, p. 45, (1936)
(20)Peyches, I. J. Soc. Glass Tech., vol. 36, pp. 164-180
(21)Wiechert, E., Wied. Ann. Physik, vol. 50, pp. 335, 546, (1993)
77
spectrum of relaxation times to explain the shape of the creep curves.
The reverse process of obtaining the relaxation spectrum from the creep
curves was attempted by Simha (22) by the application of fourier transforms.
Orwn(23)
Orowan (
derived an expression for the time independent constant
of the process from equilibrium considerations by using an analysis very
similar to that of Eyring on viscosity. He assumed, however, that the
time function can be given by an exponential much like monomolecular
reactions.
Recent experimental investigations by Jones, 4Murgatroyd and
k(25)
Pasn(26)
Sykes,
and Pearson
have demonstrated that the total amount of
elastic creep is linear with the applied stress, and proportional to a
positive power of temperature. As for the time law, empirical relations
ranging from logarithmic to exponentials of the square root of time
were suggested.
Internal dampir
which is another manifestation of the effect of
configurational order was investigated by Rotger(27) who was able to
establish from considerations of activation energies that the process
chiefly responsible for internal damping and the elstic-after-effect was
(22)Simha, R., J. Appl. Physics, vol. 13, pp. 201-207,
(23)Orowan, E., Proceed. First Nat. Cong. Ap.
(1942)
Mech., p. 458. (1951)
(24)Jones, G. 0., J. Soc. Glass Tech., vol. 28, pp. 432-462,
(1944)
(25)Murgatroyd, J. B., R. F. R. Sykes, J. Soc. Glass Tech., vol. 31, pp. 17-35,
(1947)
Pearson, S., J. Sbc. Glass Tech., vol. 36, pp. 105-114, (1952)
(27) Rotger, H. von, Glastechnisthe. Berichte, vol. 19, p. 192, (1941)
a migration of sodium ions. Fitzgerald,28) Kirby(29) et. al. have confirmed
and ellaborated on the earlier results of R8tger.
(28 )Fitzgerald, J. V., K. M. Laing, and G. S. Bachman, J. Soc. Glass Tech.,
vol. 36, pp. 90-104, (1952)
(29)Kirby, P. L. ,
.Soc.
Glass Tech., vol. 37, p. 7,
(1953)
CHAPTER
THEORETICAL
TERATMENT
I.
OF
TIME
IX
THE
ELASTIC
AFTER
EFFECT
LAWS
The following analysis is similar to that of Smith(30) and also
has many things in common with that of Eyring(31) on viscosity.
It is assumed that the strain is contributed by network modifiers
like alkali ions which have two alternate positions equilibrium seperated
by an energy barrier. See Fig. 49a and 49b. It is further assumed, a) that
the ions with alternate positions of equilibrium are uniformly distributed
and that their density is M per unit volume, b) that the barrier heights
or the activation energies i
may not be the same for every ion and that
a certain unique distribution function exists. This distribution function is
f
f (( , t
)
;
(10)
where t = time
C= shear stress.
In a virgin glass the equilibrium configuration of f is f = f(
0
In a virgin glass one may assume that half the ions are in a favorable and
the other half in an unfavorable position to contribute to the strain.
(30)Smith, C. L., Prc. Phys. Soc., vol. 61, pp. 201-205, (1948)
(31)
Eyring, H., J. Chem. Phyp.,
vol. 4,
p. 283, (1936)
DISTANCE
Fig. 49a
Free energy diagram for ion in virgin
glass
DISTANCE
Fig. 49b
Free energy diagram for ion after stress
has been applied
/ and
The number of ions between activation energies
+
in the
favorable Position is
dM = Nf = ff d
.e
Those in the unfavorable configuration are,
dM
=N
=f
df
.
At any time under any stress,
Nf (()
+ N(()
=
N(47)
=
dM(()
(11)
where N is the total number of ions in the activation energy interval d Vj.
For temperatures sufficiently below the transformation region where
stabilization processes are negligibly slow N can safely be assumed as
a constant. The number N would however be greatly dependent upon the past
cooling history of the glass, i.e. whether it had been rapidly cooled
from the molten state or stabilized for a long time at a temperature in
the transformation region.
In the course of the development three different distribution functions for f will be considered.
When the glass is subjected to a certain stress
'C the initially
symetric free energy diagram of Fig. 49a becomes antisymetric as in Fig. 49b,
resulting in a migration of ions from the high energy trough into the more
stable low energy trough. The probability for an ion in the high energy
trough to jump the barrier into an unfavorable position is given by the
kinetic theory as exp(-(JP- E)/kT).
The probability for a reverse jump is
by similar consideration exp(-(Cf+ .)/kT). The rate of strain can now be
written as
(0C1
dr
ra
where
=
ipik
(r
Jt
(12)
the strain
atomic strain contributed by one ion during one jump
=
the vibrational frequency of the ion
enery change in height of the troughs due to an
=
application of shear stress
=
diameter of an ion
k = Bolzman's constant
As the glass strains the shape of the equilibrium configuration
changes. The change in the distribution function can be written as a total
differential
(a)
yt
diL
Since
d
dt
-
dt
-
Ic4
(b)
0 during a creep test,
(c)
In view of the fact that the number of ions contributing to the
strain remains constant, it follows then that for an activation energy
interval
r y
-
/1±i W***;IA
06
or
C0-0
Vf
A'-
{. (
-I
e
vf C4
.0-V
kr/4zvfre
(13)
kT
(13a)
Further simplification gives,
*+ (2VrcA)f
- 2-V
f
(13b)
This is a first order non-hoogeneous differential equation in f the
solution of which is given in the form,
Jle
4CA
Cos4
T
= f ( Y) at time t = 0, from which
with the initial condition of f
4
s.LWe
Cos
VT
e
Simplification of (12) will result in
YCes
-
LOSL -E)t7
(14)
'a
-
e kTe
T (p
(12a)
e idye
(1 2b)
0
if now equations (12a) and (14) are combined,
4
0
Equation (12b) is the fundamental relation for the delayed elastic
strain given in terms of the unique equilibrium activation energy spectrum
of the virgin glass.
Pq
84
If it is assumed that there is only one activation energy characterizing the process, f can be expressed in terms of the delta function,
so that f, = NS(-(f-).
dC
_ydi
Then,
(2-vte kT- C.4,',.4
4~A
Z-yQ
c 2ntra
on
-A 0
-vte kT-r Los
-
P
JcT
By direct integration,
icr
~2 Y4e kT un~L~
kT
with the initial condition of f = 0 when t=0. So that,
or,
- 2,
A.k
2
In nearly every case E <kT; therefore,
k~
kT
z kT
-( "
AT
0s
tc.
(15)
kT
This relation is identical with that derived by Orowan32) It predicts
that the asymptotic value of creep will increase with decreasing temperature.
The reason for this is that the equilibrium value of the number of ions
(32)
iown, E.2 Loc
cit.
ending up in the unfavorable configuration after all transients have
died out will be larger for lower temperatures, as can be seen directly
from (14). So far all observations of the elastic-after-effect at low
temperatures have shown that the "asymptotic value" of the creep is
related to a positive power of the temperature, and that the time law of
the deformation is not exponential. Since there is no reason that the
equilibrium consideration for the number of positions should be wrong,
the appearent anomaly must be explained by the very great dependence of
the deformation rate on temperature. The very large activation energies
for the process make the relaxation times of the order of 103 to 104 years
at low temperatures. This means that after the ions with small activation
energies have been used up the rate of further deformation dies to such
low values to give the impression that the asymptotic value has been reached.
The fact that the time law is not exponential, however, gives an indication
that the consideration of a single activation energy is an over simplification.
If instead of a single activation energy it is assumed that there is
an equal number of ions wi*h each activation energy, f would then be a
constant independent of
(f
,
and can be taken out of the integral sign, so
that the integration is readily carried through.
_4 2
-r
YNkT
e4p
et 1CT
(12c)
After the very early parts of test the exponential in brackets
rapidly approaches zero, giving,
d . r AkT
cdt
4
k-T
This expression can again be directly integrated to give the strain
as,
r=
a kT
4a
kT
L t 4 C
(16)
Obviously (16) is now only good for times at which the exponential
term can be disregarded. It can be noted that the hyperbolic function in
the exponent is very nearly unity, and since the vibrational frequency of
the ions is of the order 101 per sec., the logarithmic relation ought to
be good for times larger than 10
seconds. The constant
C can be obtained
from experiments.
Although this time law is another extreme (it has no asymptote),
it can be noted that for first order quantities the strain is independent
of temperature within the region of interest.
The experimentally observed time law is always initially faster than,
and in the later stages slower than an exponential function, and lies
somewhere between the exponential and logarithmic laws. It exhibits a very
large initial slope unlike a simple exponential. These considerations
would indicate that the activation energy spectrum must be approximately
bell shaped, (perhaps a Gaussian distribution). If any other curve than
the previous two is considered the integration can only be carried out
approximately. For an approximate solution any distribution function may
be replaced by a series of rectangles. Then, the first integration may be
replaced bya sum of the form
4cl
e-
I(17)
Although the above series can be integrated further if each term
is expanded in infinite series, certain interesting trends can be observed
by inspecting the above relation. The
t in the denominator accelerates
the process in its early stages and retards it in later stages in comparison
to a simple exponential. For large time 1/t has a very small slope and the
process goes like the exponential giving a horizontal asymptote. Furthermore,
for first order quantities the strain rate is again independent of
temperature within the time interval of a regular test.
88
II. DETERMINATION
SPECTRUM FROM THE
OF THE
ACTIVATION ENERGY
CREEP CURVES
DIRECTLY
Due to the very long relaxation times for stabilization of
configurational order, the structure of glasses can be considered as
"stable" at low temperatures for the duration of normal length experiments.
Because of this it is safe to assume that the equilibrium form of the
activation energy spectrum will not change appreciably. In this case
it becomes possible to obtain the activation energy spectrum directly
from a constant temperature, constant stress creep curve by processes of
inversion.
(33)
The inversion method of Simha,
although theoretically attractive,
is in practice quite cumbersome since obtaining the fourier transforms
is as difficult as outright integration of equation (12b). Therefore, a
semi direct method will be advanced in the following paragraphs which
requires only the replacement of the activation spectrum by a series of
uniformly displaced step functions.
From part I. of this chapter,
0
Considering the following change of notation,
(33)Simha,
R., Le.
cit.
24rY &(41
V( M
ACT)
2V Go
4
a
-L
dt
equation (12b) is simlified into
t)
F(+)'
e
6
e
kT
'd
(12d)
For an approximate solution f0 ((f) can be replaced by an infinite
series of step functions each of which is displaced with respect to the
other by a constant increment of the activation energy A(fas follows:
where C is the height of the step function. Furthermore,
replaced by i A(f, resulting in
-
'ei can be
0_
(
e
where now AM(is the constant activation energy increment. Substitution of
(18a) in (12d) gives,
Fa)
d rjeekT
d(e (19)
In reality the shape of the spectrum will be similar to a Gaussian
distribution in which case it would be safe to assume uniform convergence
of the series. Now the summation and integration processes can be interchanged, giving,
1
ZO
1
M
-46P
r e~)
OCT7
V_C
CiT0(O
(19a)
90
Because of the nature of the delta function, the above relation can
be expressed more simply as follows:
C"6,)
cefI eI
/Cd
&r
TT
(19b)
Direct integration of (19b) gives
F~ 4dJ
-at'e
v.
*J~
kT
(20)
As 6(0+ Othe infinite sum approaches the integral. From physical
considerations it is clear that the high energy end of the activation
spectrum will approach zero very rapidly beyond a certain value of
Therefore, it is possible to replace the infinite sum by a finite sum.
This gives
F({U
N
A 21c~
4
where now
o
-
I
~ kT
'e(21)
A- YakT (
CT
If the experimentally observed strain rate is equated to the right
side of (21) at n different points, n linear equations in with n
unknowns in the C s will result. These equation can then be solved
simultaneously. Since different parts of the activation energy spectrum
are active at different times, the correct determination of the C. s
requires the complete creep curve starting from zero time
and extending
to very large times. Unfortunately such a perfect creep curve was not available to perform this inversion. Instead of this method a simpler but
perhaps less accurate method for the determination of the spectrum was
tried which will be explained in later chapters.
91
III.
RECOVERY
CREEP
Hitherto it was always taken for granted that the time laws of
elastic creep under stress and recovery creep were the same. Although the
difference between the two laws is slight, and perhaps in most cases
undetectable, there is nevertheless a fundamental difference.
When the stress is removed the free energy diagram goes from its
distorted state of Fig. 49b back to its symetric state of Fig. 49a. Then
the equation for the change of the spectrum becomes
do 4
t
o tm
,(22)
It is to be noted here that for recovery creep the notation of
favorqble and unfavorable is reversed.
Using (22), the equation for the strain rate can similarly be
derived as
~r= 2Y~1~4444~-..' yte
f=2
d{{
a. V
kT
j
)e
IcT
e
kTdr
(23)
A comparison of (23) with (12b) shows that there is indeed a
difference in the time law. Generally, however, as was pointed out previously,
e/kT is a very small quantity, so that the hyperbolic functions can be
replaced by their first order terms, in which case the two equations
become identical.
CHAPTER X
DESCRIPTION
I.
THE
OF THE APPARATUS
CREEP
APPARATUS
Requirements: The creep apparatus was designed for carrying out
constant stress creep tests in torsion on tubular filaments. The reason
for preferring torsion tests was as follows: a) there are no first order
dimensional changes in the specimen as a result of straining, b) the stress
can be reversed, c) the elastic-after-effect can be amplified mechanically
by using very thin specimens.
It was decided to have the specimens in the shape of long thin
tubes, because, a) this would assure a nearly uniform distribution of
stress, b) residual temper stresses could be eliminated, c) and most
important, the configurational order as a function of cooling rate could
be controlled with ease over a large range.
It was also decided that the creep should be recorded.
Design information: A scematic view of the resulting instrument can
be seen in Fig. 50. Nearly constant stress was achieved by a 0.004" D.
tungsten filament which was 100 inches long (1).*
(*)Numbers refer to those in the diagram.
The relaxation of torque
15
Fig. 50 Schematic view of creep device
94
in the filament due to the delayed elastic and viscous deformations in
the glass specimen in an average experiment was of the order of one per
cent of the applied torque. Since the tungsten filament is the most
highly stressed member in the whole system, joining it to the other members
presented some difficulties. The problem was finally resolved by bending
the two ends of the filament at 900 angles to form continuous levers, and
brazing these lever arms between two flat copper plates which were then
joined to the rest of the system. The section connecting the tungsten
filament to the specimen was a hypodermic tubing of 19 gauge thickness.
This tubing carried the two front surface mirrors (2) placed back to back,
and the deflection pointer (4). Two vanes which were attached to the
pointer were immersed into an annular oil bath (5) to serve as a vibration
damper. The lower end of the hypodermic tubing carried an elastic chuck to
hold the specimen (6). Another elastic chuck with two diametrically
opposed notches was suspended from the lower end of the specimen. A fork (14)
which engaged into the notches of the lower chuck served for winding the
specimen. The total angle of twist could be measured on a goniometer (10)
to which the lower end (9) was integrally attached. The'whole system of
torsion filament, connecting hypodermic tubing, specimen, etc. was encassed
into a vacuum system to eliminate the disturbing effects of convection
currents generated by the furnace (7) during the high temperature tests.
The stopcock opening (8) was provided for sucking through liquid nitrogen
vapor for rapid cooling. Functions of this rapid cooling technique will be
ellaborated in later chapters.
Cf. post.
p.
I
95
The vacuum was broken at the special valve (15). The level of the
oil in the vibration damper could be adjusted by raising or lowering the
attached glass bulb (11). This system also served as a rough vacuum gauge.
The backbone of the apparatus was a drill press base and column (12). The
weight of the table plus that of the furnace was counterbalanced by a
weight suspended inside the hollow column.
The furnace tube was covered with a sandwich of asbestos-aluminum foilasbestos to damp out the temperature waves resulting from the intermittent
heating action of the furnace. A chromel-alumel thermocouple placed
between the damping sandwich and the furnace heating coils was the temperature
sensing element. The temperature was controlled by a Brown "Electronik"
indicating controller with a micro switch within one degree Celcius.
Specimens were inserted into the system by disconnecting the special
joint (13) and swinging the table carrying the furnace aside, so that the
furnace tube would become directly accesible.
The stress was applied by twisting the lower end piece (9) around
until the upper pointer (4) went through the desired number of turns.
The difference between the readings of the lower goniometer (10) and the
upper pointer is the twist of the specimen (not necessarily all elastic).
While winding the assembly the level of the oil in the damper was lowered
so that the specimen would not be over stressed by the viscous drag of the
damper.
The strain recording arrangement consisted of a 2 watt Sylvania
concentrated arc lamp (1), Fig. 51, a high angular magnification reading
17
VI
Fig. 51
Recording system
A
telescope (2), and a recording drum carrying photosensitive paper (3).
The drum revolved at a rate of one revolution per hour and was driven
by a syncronous clock motor. The light generated by the concentrated arc
lamp was directed by the telescope unto the front surface mirror (2),(Fig. 47)
of the creep device and the reflected rays were focused a$ a point on the
camera drum standing next to the light source and the telescope. Thus, the
rotation of the mirror as the specimen strained was recorded continuously
on the camera drum. The recording apparatus was properly enclosed in a
tent to eliminate clouding of the record.
A side view of the instrument is seen in Fig. 52.
Calibration 6f
jhe
torsion spring: The tungsten wire torsion spring
was calibrated by attaching a body of known moment of inertia on the
hypodermic tubing with the mirror and the pointer removed, and determining
the frequency of the free oscilations. From such a test the stiffness of
the torsion spring was obtained as
k = 6.7038 x 10-3
II.
grf - cm/rad
TIE SPECIMENS
Method of preparation of specimens: The raw material for the specimens
was the #7740 pyrex glass of Corning with an average composition of
Sio2 80%, B203 145, Na20 4%, and Al0
2 . This glass was readily obtainable
in 1.5 mm. diameter tubes of 12 inch lengths under the name of melting
point tubes. The specimens were prepared by softening a small section in
"U'
~1
Fig. 52 Side view of creep apparatus
99
the center on a flame and drawing the diameter down to the desired size.
After lon hours of tedious hand drawing, it was decided that specimens
of sufficient uniformity, symmetry, and thinness in wall could not be
obtained by hand. A mechanical device was then constructed which was
capable of producing more uniform, symnetric, and thin walled tubes. The
device can be seen iji Figs. 53 and 54. The glass tube (1) is held from
both ends by elastic chucks (2). The two grips are rotated at the same
speed by a gear arrangement while a small section in the center is
softened by a hot flame with-a sharp point. When the heated sectin of the
tube assumed an orange color, the flame was removed and the driwing motor
was abruptly stopped. The sudden deceleration disengaged the shafts from
the driwing collors, and the compressed springs threw the shafts out a
prescribed distance until they were stopped by plasticine covered bumpers.
To obtain thin walls at the test section the glass tubes were sealed at
both ends before drawing. When the center portion was heated during drawing,
a slight air pressure inside the tube prevented the thickening and
gradual collapse of the wall. In this fashion wall thickness to radius
ratios in the neighborhood of 1/10 were easilt obtained. Fig. 53 shows the
device in the compressed and Fig. 54 in the extended condition. Two such
drawn filaments can be seen in Fig
.
55 while Fig. 56 shows a view of a
typical tube under the microscope. It can be seen that the wall thickness
is but a few microns.
Inspection and measurement of specimens: The uniformity of the
specimens was checked roughly after drawing by bending the central section.
As the bent section was moved from end to end the change in the radius of
100
Fig. 53
Filament drawing device in compressed position
Fig. 54 Filament drawing device in extended position
101
~j~*:
Fig. 56
Photomicrograph of typical tube
Fig. 55
Tubular torsion creep specimens
102
curvature was noted. Only specimens with very small radius of curvature
were accepted. Despite every precaution, and extreme care, however,
specimens could be brought only within 10% deviation on the diameter.
The final measurements of the inside and outside diameters were made
within a few microns accuracy under a large metallograph, see Fig. 56.
Diameter measurements were made at points one centimeter apart all along
the gauge length. Since the diameter of the specimens decreased gradually
from the parent 1.5 mm size to the diameter of the test section, the
question of what to call the gauge length was settled by plotting the
area moment of inertia of a typical specimen from the value at the
uniform test end to the uniform large end over the varying conical portion,
as seen in Fig. 57. From this plot it was decided that the gauge length
should be taken 2.2 centimeters less than the shoulder to shoulder distance.
103
DISTANCE
Fig. 57
FROM
SHOULDER ,
CM.
Contribution to the angle of twist of the ends of the
specimens
CHAPTER XI
EXPERYIENTAL RESULTS
I. GENERAL
DESCRIPTION OF TESTS
The primary objectives of the tests were to a) check the temperature
dependence of the asymptotic value of the elastic-after-effect and the
viscosity, b) determine the activation energy spectrum for the process.
In view of the fact that the geometry of the specimens as well as
the composition (*) at the test section could not be reproduced, all the
tests were carried out with the same specimen. The outside and inside
diameters of the specimen were measured under a microscope at one centimeter intervals all along its length.
Since it is impossible to wait until the kinetic processes die out,
the tests at low temperature were basically of a short time nature. A
typical test history was as follows: The test specimen was stabilized
for ten hours at 5100 C in a seperate furnace, it was then cooled to room
temperature and was mounted into the creep apparatus. After the temperature
of the furnace came to an equilibrium, the level of the oil in the damping
basin (5),
Fig. 50, was lowered leaving the filament and specimen system
Although great care was taken in drawing the specimens quickly before
the flame showed any yellow coloring, the variation in creep between
seemingly identical specimens was large.
105
entirely free. The system was then wound up by turning the lower end (9) around
to give the upper pointer (4) the desired number of revolutions. Care was
taken not to over stress the specimen in any way by accidental over
winding or by excessive vibrations. After the winding was completed, the
level of the oil in the damping basin (5) was raised to damp out the
unavoidable vibrations. The final adjustment to bring the light on the
desired place of the camera drum was made by turning the lower end (9)
around slightly. The total angle of twist from the start to this moment
was recorded from the goniometer (10). A seperate stop watch was used
for the determination of the time reference. Since the process starts as
soon as winding commences, the stop watch was started simultaneously with
the winding. After the light beam was brought on to the camera drum and
recording began, a time marking was put on the record by intersecting
the light beam while simultaneously stopping the watch. This gave the
time reference on the record. Because of the time required for winding the
specimen, the initial portions of the creep curves were never captured.
The problem of seperating the elastic deformation from the elastic-aftereffect in the goniometer reading was resolved as follows. After the
twelve hour test under stress the heating was stopped, and the specimen
was cooled under stress as rapidly as possible to a temperature in the
vicinity of -100
C to freeze nearly all thermally activated processes.
This was achieved by removing the furnace (7) from around the furnace tube,
breaking the vacuum and using the vacuum pump now to suck through opening
(8) liquid nitrogen vapor which cooled the specimen to a temperature in
the vicinity of -1000 C. The anamolous deformations due to the variation
106
of elastic modulus with temperature were eliminated by moving the light
back to the same point on the recording camera during the process of
cooling. When the specimen was now unwound at this low temperature, the
thermally excited processes were frozen in so that the deformation
immediately recovered upon unloading was the elastic part. After the
elastic part was established, the temperature was quickly raised to the
test temperature and normal recovery proceeded, as usual. This constituted
the twelve hour recovery test. The viscous part of the creep under stress
was seperated by plotting the creep curves under stress and recovery creep
and subtarcting the latter from the former. The viscous deformation was
then subtracted from the creep curve under stress to determine the delayed
elastic deformation.
A ten hour annealing or stabilization test at 5100 C between tests
assured a virgin and a nearly identical structure for each test. Since this
temperature is 50 C below the strain point and because the nature of the
tests, no first order changes occured in the geometry of the specimen.
In this fashion the errors due to measurement and reproducibility of
structure were kept constant in the first case and eliminated in the
second case.
II. EFFECT
OF
TEMPERATURE
ON THE DEFORMATION
The theory of the elastic-after-effect(35) predicts that the asymptotic
value of the total delayed elastic strain is inversely related to tempera-
(35)Cf. ante. p.64
1
107
01
x
NIVHILS
COiSVw1-
03AV-130
Fig. 58 Delayed elastic creep at various temperatures
108
20
10
19
10
18
10
175
200
225
TEMPERATURE,
250
275
*C
Fig. 59 Variation of viscosity with temperature
300
109
ture. Unfortunately this could not be varified with the twelve hour
experiments at low temperature.
The specimen whose dimensions are given in Table I was used for these
tests. The reason for choosing a range from 1750 C to 2750 C is that
short time tests at lower temperatures do not give very useful information
because most of the activation spectrum never contributes anything to
the strain. At temperatures highr than 3000 C the process is quite
rapid and now requires reseting the image of the light on the camera
drum at short intervals. The procedure of carrying out the experiments
was described in part I of this chapter.
The delayed electic deformation at these temperatures resulting
from this series of tests can be seen Fig. 58. The appearently anamolous
behavior of the curve at 5230 K has no physical meaning and probably
results from an error made in determining the zero point. A general
trend of increasing deformation with temperature can be observed.
The change of viscosity of this structure is plotted in Fig. 59.
The viscosity for the test at 1750 C could not be determined because the
recovery creep curve at 40,000 seconds still showed effects of the
initial chilling -operation. The viscosities check well with those in the
literature,(36) and are some of the highest values ever recorded.
(36 )Pearson, S. Ibid. p. 112
Ti
110
TABLE
DIENSIONS
I
OF SPECIVEN
1
Outside Diameter
microns
Inside Diameter
microns
182
159
133
116
123
106
118
100
118
103
118
103
123
106
106
91
110
94
112
97
109
94
106
91
106
91
106
93
112
96
118
103
137
118
206
178
111
III.
DETERMINATION
OF
THE
ACTIVATION ENERGY SPECTRUM
Since the inversion process described in Chapter IX could not be
performed from lack of a proper creep curve, a different, more direct
experimental method had to be resorted to.
The method consists of running a high temperature (3000 C ) creep test
as usual, at the end of 24 hours cooling the specimen -1000 C and
untwisting it at that temperature, then permitting it to recover at
steps of increasingly higher temperatures. The temperature was increased
one step only after the rate of recovery at the lower temperature had
become very small. In this fashion the recovery spectrum was eaten away
step by step, the amount of recovery taking place at each step being
an indication of the height of the corresponding column in the activation
energy spectrum.
A new specimen the dimensions of which are given in Table II was
used for these tests. As usual the specimen had been stabilized at 5100 C
for ten hours prior to testing. The creep at 3000 C lasted for 24 hours.
The recovery time at different temperatures
after unwinding at - 10 0 0 C
is given in Table III. The resulting recovery creep curves can be seen
in Fig. 60. By a simple process of conversion (37) the qctivation energy
spectrum
Cf. post
can be obtained from the results of Fig. 60. This spectrum
p. 117
(*)Or the part of the spectrum that has contributed to the creep test
under stress.
112
0
2
4
6
8
10
012
14
16
18
20
6/3~ K
22
RECOVERY
60000
40000
20000
TIME
,
SECONC
2
Fig. 60 Step by step recovery of a pre strained specimen
113
TABIE
DIlNSIONS
II
OF SPECIFEN
2
Outside Diameter
microns
Inside Diameter
microns
194
160
139
118
124
100
116
97
112
94
112
94
112
93
105
88
103
86
108
88
110
92
109
91
109
90
112
94
118
98
132
110
188
159
114
can be seen in Fig. 61.
TABLE
Temperature
III
Time at Temperature
seconds
Converted Absolute
Recovery Time
24"0
43,680
1000 C
44,520
1750 C
41,340
41,387
2500 C
86,820
86,933
3250 C
89,160
89,610
4000 C
59,820
60,654
"t
43,680
seconds
44,521
I"
115
-5
8x10
-5
6x10
-5
4x10
-5
2 x 10
20
ACTIVATION
Fig. 61
30
ENERGY,
50
40
K-CAL / MOLE
Activation energy spectrum as determined from Fig. 60
60
CHAPTER XII
DISCUSSION OF RESULTS
I. THE
TEMPERATURE
DEPENDENCE
TESTS
As was repeatedly mentioned before, the high temperature tests
did not show the predicted fall in the "asymptotic value" of the elastieafter-effect primarily because in short time tests the asymptotic value
is never reached. To give the theory a fair test, creep tests should be
performed at a high enough temperature where relaxation times are short
enough to make it feasible to wait until the process dies out, and yet
at a low enough temperature where stabilization times for the establishment
for the equilibrium configuration are much longer than the test period.
A temperature range about 500 C below the transformation region may
be the answer to this.
The viscosities observed from these tests are of the order of
18
19
10
- 10
poises. The very rapid rise of the viscosity with dropping
temperature is characteristic of a strongly associated liquid like glass.
In this temperature range and lower for all intents and purposes glass can
be considered as a rigid elastic solid. Its viscous properties contribute
negligibly little to its behavior under moderate engineering stresses.
Manifestations of viscosity can only be observed by highly sensitive
special apparatus or under enormously large stresses like those under very
sharp pointed diamond indenters.
117
II.
ACTIVATION
ENERGY
SPECTRUM
The activation energy spectrum was calculated from the recovery
creep curves of Fig. 60 as follows:
The general equation for recovery creep is
drw 2
-
df
elc(P~
V01
e-..k.F
kT
If the equilibrium spectrum f
(
(23)
is written in terms of the
spectrum which is potentially contributive to strain §() at a given
temperature and stress, where the latter can be defined as
) (t 4-(('0)- 4,y
(*)
then, the equation (23) is simplified to
"r-
.5
2
((p)
e
e T-d~
(23a)
0
Integration with respect to time gives
YAt
(24)
0
The term in brackets
- z-vfeEkT
f
f
eq
)
is the final shape of the spectrum at infinite time for the
favorably oriented ions.
118
is termed the eating function because of its characteristic operation
on the activation spectrum. For a given activation energy
C
,
the function
S goes from zero to unity as time progresses. Moreover, the rise in the
function S is more rapid for lower activation energies which is synanemous
with saying that points with lower activation energies have shorter
relaxation times. From this one can see that the function S is very nearly
a unit step function which advances in the direction of high activation
energies as time advances. See Fig. 62. For a certain temperature the
rate of advancement slows down at a certain activation energy, and
progresses increasingly slower from there on. At this time the function
S will have eaten away the portion of the activation spectrum falling
below the point where it has slowed down. See Fig. 63. Now, when the
temperature is increased, there is a new surge, and the function S
accelerates in eating away from the recovery spectrum until it is slowed
down again at a higher activation energy. This is what the integral in
equation (24) means, and the explanation for Fig. 60.
Clearly then, if the frequency for the ionic vibration 4 is known,
one may get an indication where the function S slows down in the activation
spectrum. For the purpose of interpreting Fig. 60, % was assumed to be
10 3 per second. This value was picked from Rbtger's
internal damping
results on Jena glass which is very similar to pyrex in composition.
The heights in the spectrum in Fig. 61 were then obtained by dividing
the incremental total recovery creep strain by the activation energy
(38)R8tger, H., Ibid.
p. 198
119
ACTIVATION
ENERGY
Fig. 62 The "eating function" S
-V
/
/
/
/
/
/
/
/
/
/
/
/
/
/
J
ACTIVATION
ENERGY
Fig. 63 A partially used up activation energy spectrum
120
increment, and normalizing this curve afterwards.
here that the normalized spectra
(((40)and
f 0()
It may be pointed out
are identical.
CHAPTER XIII
CONCLUSIONS
The creep apparatus proved to be well suited for carrying out
investigations on the structure of glass by means of creep and recovery
tests on tubular glass filaments.
A method was perfected for drawing the specimens and varying their
configurational order.
The general findings of this series of tests are briefly enumerated
below.
1.
The inverse effect of rise in the asymptotic value of the
delayed elastic effect could not be observed in the short
time, low temperature experiments.
2.
The viscosities at the 2000 C to 3000 C range of a structure
stabilized at 5100 C were of the order 10
3.
- 10 19 poises.
The activation energy spectrum for a 5100 C stable structure
was determined from a set of relaxation experiments which
consisted of step by step thawing of a pre strained specimen.
The peak of the spectrum appears to be at 48,000 calories per
mole.
ITi
PART
DIFFERENTIAL
III
REFPACTOVETER
(STRESS
METER)
CHAPTER
DIFFEENTIAL
XIV
REFRACTOMETER
(STRESS
METER)
The following portion of the thesis deals with the design of an
instrument for the measurement of the residual surface stress on tempered
glass. This work is a direct continuation of this investigator's Masters
degree thesis which contained an analysis of the feasibility of the
method.
I.
THE PROBLIEM
Statement of the problem: The purpose of this work was to design an
instrument which could measure the residual surface stress on tempered
glass in a rapid and non destructive manner.
Importance of the problem: The measurement of residual stress on
tempered glass resulting from the chilling operation is of great interest
to the glass industry. The method which was used hitherto consisted of
determining the relative retardation of light sent through the plate
parellel to its surface at different depths. In this fashion the stress
can be determined at every point starting from the tensile stress in the
interior to the compressive stress at the outer layers. The stress at the
(39)Argon, A. S., M. I. T. Masters Thesis in Mechanical Engineering, (1953)
124
very surface is then obtained by extrapolating the curve outward. It can
be seen that this method has serious shortcomings. It is laborious, it
is not capable of giving direct surface measurements, and worst of all
it can not be applied to a commercial plate of large dimensions, as well
as curved plates. The method of scattered light from a narrow beam was
tried by Casella and Resnick.(40) Their results were only qualitative and
of limited use. With the present instrument it is possible to make direct,
rapid, and nondestructive measurements of the surface residual stress on
nearly any size glass plate, as long as the surface is smooth. The
instrument works on the principle of critical angle refractometers, the
difference being that absolute measurements are sacrificed for high
sensitivity at a given range.
II. FUNDAMENTAL THEORY
Definition of terms used: An elementary and detailed treatment of
the fundamental concepts of light, birefringence of crystals, and birefringence of stressed media is given in this investigato's Masters thesis 41)
Theory: It is well known that many crystals with optical properties
exhibit birefringence, and that isotropic media with optical properties
become birefringent when they are subjected to stress. The anisotropy
resulting from the applied stress can be given by the following general
(40)Casella, Resnick, M. I. T. Bachelors Thesis,
(41)Argon, A. S., _p..ij.
(1951)
125
equatio ns-4. (42)
+ C(
y+G)
2y
z
n -x n0 = C
x
(25a)
n -n 0 =C
(25b)
nnz n = C1 z ++ 2(+Cr)
x+C
~ o
x)
(25c)
Where the x-y-z system of axes refers to the set of principal
stresses
', I~,
(z.
The left hand side of these equaltions gives the
the change in the refractive index resulting from the applied stresses.
C and C2 are stress optical constants.
For a large plate of tempered glass having the x-y plane as its
surface, these relations simplify in view of the fact that IT = q- and
x
y
Cf = 0 on the surface. So that,
n
- nn
=
+
nz - o = 2C2
2
x
= (C + C
x
x
(25a, b)
(25c)
If no is eliminated between (25a, b) and (25c),
n - n
(C - C2)
(26)
This indicates that it may be possible to measure the stress on the
glass by measuring its birefringence refractometrically. The highest
commercial temper stresses on glass are in the vicinity of 2 x 104 psi.,
(42)Drucker, D. C., Handbook of Experimental Stress Analysis, p. 931, (1950)
126
while the differential stress optical sensitivity for soda glass is only
2.7 brewsters, or 1.86 x 10~
/psi. Thus, the highest temper stress causes
a change in the refractive index of two units in fourth decimeal place.
The best commercial refractometers can do no better than an uncertainty
of one unit in the fifth decimal place. This would mean an error of roughly
5 per cent. It is possible, however, to design a refractometer which has
high differential sensitivity at a sacrifice of absolute measurements.
The instrument which will be ellaborated in later chapters was designed
along these lines. Other potentialities of this stress meter in the field
of classical photoelsticity will be dealt with in the concluding chapter.
CHAPTER
METHOD
OF
XV
PROCEDURE
Before the details of the design of the differential refractometer
are discussed, a brief review of critical angle refractometers will be
made.
I. CRITICAL ANGLE REFRACTOMETERS
A critical angle refractometer has two fundamental parts: a) the
refracting prism, b) the viewing telescope. The working principle of a
critical angle refractometer is illustrated in Fig. 64. Prism A has a
refractive index ND which is larger than the refractive index nD of the
body B. A bundle of monochromatic rays is incident on the contact surface
y-y. Rays having an angle of incidence less than C4 are partly reflected
at y-y, and are partly refracted into body B. Rays having an angle of
incidence larger than o< , are totally reflected at y-y. If the reflected
rays leaving surface y-y qre viewed through a telescope focused at
infinity, two regions of different brightness seperated by a sharp boundry
are seen in the field of view. This is due to the sharp change of intensity
of the reflected light at the critical angle. Here the function of a
telescope with infinity focus is to group rays of equal intensity (making
the same angle with the normal to the reflecting surface) into points on
the back focal plane of the telescope. In ordinary refractometers the
128
ANo
N
y
BP1
Fig. 64 Refracting prism of a refractometer
AIR
ny
y
B
no
Fig. 68 Refracting prism of differential refractometer
129
G
A5
A,
,"
141
bL;
Fig. 65 The Abbe Refractometer
130
telescope is attached to a goniometer, permitting a direct reading of
the critical angle, see Fig. 65. Since the refractive index of body A
is known, the index on the test body can be calculated from Snell's law
of refraction.
II. DIFFERENTIAL
REFRACTMETER
(STRESS
METER)
The stress meter, a cross sectional view of which appears in Fig. 66,
functions on the same principle as a refractometer. Here it is attempted
to increase the sensitivity of the refractometer by a suitable choice
of refracting prism.
The different parts of the instrument are as follows:
1.
100 watt mercury-arc light source
2.
Lamp housing
3.
Front surface mirror
4.
Adjustable condenser lenses
5.
Main light tube
6.
Wratten 77 filter in "A" glass
7.
Iris diaphragm
8.
Sliding carriage fer the main light tube
9.
Adjusting screw for the above carriage
10.
Leaf spring backing carriage 8
11.
Tiltable front surface entrance mirror
(*)Numbers refer to those in Fig. 66
131
90,
-0
I12)
Fig. 66a Cross section of stress meter
132
A
Fig. 66b Side view of stress meter
133
12.
Mirror tilting screw
13.
Polaroid filter with axis at 450 to surface of glass plate
14. Refracting prism
15.
Exit mirror tilting screw
16,
Tiltable exit mirror
17.
Telescope carriage adjusting screw
18.
Sliding telescope carriage
19.
Polaroid analyser
20.
Telescope objective
21.
Telescope tube
22.
Pinion wheel for focusing adjustment
23.
Eyepiece tube
24.
Screwed ring carrying reticule
25.
Reticule inserted at back focal plane
26.
Eyepiece
The general frame of the prism housing is in direct contact with
the glass to be tested. The refracting prism sits on the glass by its
own weight. In this fashion a uniform thickness of the contact liquid is
assured.
The light of the mercury-arc lamp is reflected into the main light
tube, and is filetered by a Wratten 77 filter which absorbes effectively
all the characteristic peaks of mercury vapor light except the green at
5461 R. The monochromatic light is focused on the center of the contact
surface between prism and glass by the set of three convex lenses.
134
The reflected light leaving the prism is reflected by the tiltable
front surface mirror into the analyser and telescope tube. The analyser
permits extinction of one boundry while the position of the other is read.
This procedure proved very handy in measurements on curved glass where the
quality of the surface is poor, resulting in a fuzzy image.
The distance between the two characteristis total reflection
boundries is read directly by means of a reticule. The appearence of the
field of view illustrating the stress bands for glasses with various
degrees of temper stress can be seen in Fig. 67.
The design information and the functioning of the instrument is
given in Chapter XVI.
Fig. 67 Stress bands in the field of view of the stress meter
CHAPTER
DESIGN
OF
DETAIIS
XVI
THE
STRESS
METER
The details of the design of the stress meter are handled below in
the order of their importance.
I.
REFRACTING
PRISM
The sensitivity of a refractometer to changes in refractive index
in the test material can be increased considerably by the proper choice
of the glass and geometry of the refracting prism.
Fig. 68 shows a refracting prism in contact with a glass of index n.
The refractometric sensitivity can be defined as the rate of change of
with refractive index n, at the critical angle of the surface y-y.
Sensitivity =
(27)
From Snell's law of refraction,
/
Ctox
(28)
CeO-O
Clearly, then, the sensitivity can be increased by:
a) bringing the refractive index ND close to nD so that cto V will
become very small, or,
137
b) choosing an appropriate vedge angle 0 so that
P
will be large,
giving the same result.
Although both possibilities are potentially capable of improving
the sensitivity, the first one has fewer limitations. As the refractive
index N approaches nD, the view becomes less contrasty. Below a certain
level of contrast the changes in intensity due to interference effects
of the film of the contact liquid obscure the characteristic boundry.
Furthermore, the commercial awailability of optical glasses for prisms
having indices in this range is limited. For these reasons an extra
light flint glass of refractive index ND = 1.5415, supplied by Bausch &
Lomb Co. was used for the prism. This gives a critical angle with ordinary
plate glasses of approximately 810.
From previous analysis
effect of the large exit angle
it became clear that the beneficial
could not be utilized because of the
general shape of the light intensity curve leaving the prism. When 1
approaches 900,
C< approaches the critical angle of the exit surface,
and the intensity of the refracted light leaving the prism drops sharply.
Thus, the step in the intensity curve due to the primary total reflection
at surface y-y is completely overshadowed by the general very rapid drop
of the intensity of the light leaving the prism. For this reason and for
simplicity in manufacturing, the angle
(43)Argon, A. S., Loc. _cit
0
was chosen to be 900.
0
138
II. THE
TELESCOPE
The telescope was made by combining a 29 cm. focal length front
element of a convertible Protar photographic lens and a 30x Bausch & Lomb
Balscope Sr. telescope eyepiece. A reticule of 20 lines per millimeter
was inserted at the back focal plane for direct measurement of the distance
between the characteristic total reflection boundries.
III.
OTHER
OPTICAL
PARTS
The mercury arc lamp together with its proper transformer, and the
three convex lenses were secured from Central Scientific Co. in Cambridge.
The Wratten 77 filter was purchased from Eastman Kodak Co. The large
front surface mirror (3), and the optically flat small front surface
entrance and exit mirrors were ordered from A. D. Jones Optical Co. in
Cambridge. The optically flat polaroid analyser was obtained through the
Polaroid Corporation in Cambridge.
All metallic parts of the instrument were constructed in the
Materials Division workshop at the Institute.
CHAPTER
CALIBRATION
I.
OF
PROCEDURE
THE
OF
XVII
STRESS
YETER
CALIBRATION
A special loading device was constructed fro the direct calibration
of the stress meter. A prism of optical crown glass of refractive index
nD = 1.523 and having dimensions 1 x 2 x 3 inches was used as the specimen.
One of the 2 x 3 inch surfaces was optically flat within one wave length
of light. Measurements with the stress meter were made from this side.
Fig. 69 shows the calibration set up consisting of loading device (1),
test block (2), stress meter (3).
The test piece was compressed across its 1" x 2" face in a vertical
compression device in a polariscope to determine the stress optic constant
of the prism. The calibrated prism was then compressed in a special
compression device shown in Fig. 69 in a horizontal position bearing the
stress meter on the optically flat surface. Thus, it was possible to
observe the load on the prism and the birefringence of the surface
simultaneously. Bending of the prism was eliminated by maintaining the
stress pattern in the polariscope symmetrical. The resulting calibration
curve can be seen in Fig. 70.
From the exact dimensions of the cross sectional area (0.996" x 1.944"),
the stress constant of the stress meter is evaluated as
140
r Z,,u
4
/
an
t0
'01h,
Fig. 69 Calibration set up for stress meter
141
0
2
EYEPIECE
4
6
MICROMETER
8
READING
Fig. 70 Calibration curve of stress meter
10
142
k = 946 + 50 psi/ eyepiece division
The measurement of the stress band in the eyepiece field was made
by extinguishing one boundry while reading the position of the other. It
was found that this method gives more reliable and reproducible results.
II. ACCURACY
It is impossible to read the positions of the boundries of the
divided field closer than one half of one division, corresponding to
about 500 psi., i.e. about 2% of commercial temper even when the glass
surface on which the measurement is made is optically flat. This
limitation on the accuracy results from the fact that the boundry is
not a sharp one but instead a series of alternating dark and light bands
characteristic to Frauenhoffer type diffraction phenomena, see Fig. 67.
Since in most cases the diffraction bands do not appear as a result of
poor surface conditions, the boundry has a fuzzy appearence. As would be
expected the quality of the view is more severely influenced by surface
striations which could hardly be seen with the naked eye, than by large
gradual waviness.
Measurements on wavy surfaces showed that the error of measurement
is the same as for an optically polished surface, i.e. 1/2 scale division,
if the surface striation is parallel to the optical plane of the instrument.
For transverse surface waves, however, the error is increased to one
division and occasionally to higher values.
143
A difference between the base refractive index of the calibration
prism and the glass to be tested would constitute another source of error.
If the bulk refractive index of the test glass is known, a correction
factor to be multiplied with the stress constant can be used to eliminate
this error.
It can be shown that this factor is
2
nD
N
no
2
ND
1
- n
2
2
nD
where N = 1.5415, refractive index of the prism of the stress meter
n
= 1.5230, refractive index of the calibration prism
nD= bulk refractive index of the test glass
CHAPTER
XVIII
CONCLUSIONS
The stress meter which has been described is capable of rapid, nondestructive determination of residual surface stress on tempered glass,
with little limitations as regards form. The accuracy of measurement of
high degrees of temper under normal conditions is 2-3%. This only a small
gain over the power of a first grade commercial refractometer.
The main
advantage of this instrument is then, more in the direction of the direct
applicability of the refractometric method with a simple design to the
problem of measurement of residual surface stresses on glass. The method
is potentially appliacable to non transparent but reflecting glassy
surface layers.
A very promising application in the field of classical photoelasticity
can also be envisioned. The stress meter reads the stress in the surface
layer acting across its optical plane. Since the refractive index is
not effected by shear stresses but only by normal stresses that produce
dilation, in a two dimensional stress field like that encountered in
photoelastic models the stress meter will read directly the magnitude of
the normal stress acting across the optical plane. By rotating the
instrument around the surface normal passing through the point of interest
on the photoelastic model, it is possible to determine the normal stresses
at this point in several different directions . Then by the use of a Mohr
145
circle construction akin to strain rosette methods it would be possible
to determine the complete stress picture at a point. This method of
determination of stresses will obviously never take the place of a polariscope
picture, but it may be at least a powerfull tool to suplement it.
BIBLIOGRAPHY
Argon, A. S., "A New Optical Method for the Measurement of Residual
Stress in Tempered Glass", M. I. T. Masters Degree Thesis, (1953)
Auerbach, F., Ann. Physik. Chem., vol. 53, p. 1000, (1894)
Bachman, G. S., see Fitzgerald, J. V., K. M. Laing, and G. S. Bachman
Bernal, J. D., Transactions of the Faraday Society, vol. 33, p. 27, (1937)
Casella & Resnick, Application of Narrow Light Beams to the Photoelastic
Stdy of Tempered Glass, M. I. T. Bachelor's Thesis, (1951)
Douglas, R. W., "Relation Between Physical Properties and the Structure
of Glass" Parts I and II, Journal of the Society of Glass
Technology, vol. 31, pp. 50-89, (1947)
Drucker, D. C., Handbook of Experimental Stress Analysis, M. Hetenyi editor,
p. 931, Wiley (1950)
Eyring, H.,
Journl of Chem. Physics, vol. 4, p. 283, (1936)
Fisher, J. C., J. H. Hollomon, Metals Technology, vol. 14, T. P. 2218, (1947)
Fitzgerald, J. V., K. M. Laing, and G. S. Bachman, "Temperature Variation
of the Elastic Moduli of Glass", Journal of the Society of Glass
Technology, vol. 36, pp. 90-104, (1952)
147
Kinetic Theory of Liquids, Oxford University Press, (1946)
Frenkel, J.,
Hertz, H.,
Gesanmelte Werge. vol. 1, p. 155, (1895)
Hollomon, J. H., see Fisher, J. C. and J. H. Hollomon
Huber, H., Annalen der Physik, vol. 14, p. 153, (1904)
Jones, G. 0., "Viscosity and Related Properties in Glass", Reports on
Progress in Physics, vol. 9, p. 136, (1948)
"Determination of the Elastic Viscous Properties of Glass at
Temperatures Below the Annealing Range", Journal of the Society
of Glass Technology, vol. 28, pp. 432-462, (1944)
Kirby, P. L.,
Journal of the Society of Glass Technology, vol. 37, p. 7
(1953)
Kohlrausch, F.,
"lUeber die Elastische Nachwirkung bei der Torsion",
Pogg. Ann. Physik, (4), vol. 29, p. 337, (1863)
"Beitrdge zur Kentniss der Elastischen Nachwirkung 2 , Pogg. Ann.
Physik, (5), vol. 8, pp. 1, 207, 399, (1866)
"Experimental Untersuchung fiber die Elastische Nachwirkung bei
der Torsion Ausdehnung und Biegung", Pogg. An. Physik, (6),
vol. 8, pp. 337, (1876)
Kohlrausch, R., "Theorie der Elektrischen Rtickstandes in der Leidener Flasche",
Pogg. Ann. Physik., (4), vol. 1, pp. 56, 179, (1854)
148
"Nachtrag tiber die Elastische Nachwirkung beim Cocon, und
Glasfaden etc", Po.g. Ann. Physik, (3) vol. 12, p. 393,
Krutter, H.,
(1847)
see Warren, B. E., H. Krutter, and 0. Morningstar
Laing, K. M., see Fitzgerald, J. V., K. M. Laing, and G. S. Bachman
Lillie, H. R.,
Journal of the American Ceramic Society, vol. 14, p. 502 (1931)
Journal of the American Ceramic Society, vol. 16, p. 619,(1933)
Journal of the American Ceramic Society ,vol. 17, p. 43, (1934)
Journal of the American Ceramic Society, vol. 19, p. 45, (1936)
Morningstar. 0., see Warren, B. E. , H. Krutter, and 0. Morningstar
Murgatroyd, J. B., R. F. R. Sykes, "The Delayed Elastic Effect in Silicate
Glasses at Room Temperature", Journal 'of the Society of Glass
Technology, vol. 31, pp. 17-35, (1947)
Orowan, E., "Creep in Metallic and Non Metallic Materials", Proceedings of
the First National Congress on Applied Mechanics, pp. 453-472,
(1951)
Pearson, S., "Creep and Recovery of a Mineral Glass at Normal Temperatures",
Journal of the Society of Glass Technology, vol. 36, pp. 105-114,
(1952)
Peyches, I., "The Viscous Flow of Glass at Low Temperatures", Journal of the
Society of Glass Technology, vol 36, pp. 164-180, (1952)
149
Powell, H. E., F. W. Preston, Journal of the American Ceramic Society, vol. 28,
pp. 145-149, (1945)
Preston, F. W., see Powell, H. E. and F. W. Preston
Resnick,
Casella and Resnick
see
Rtger, H. von, "Elastische Nachwirkung durch Wgrmediffusion und Materiediffusion", Glastechnische Berichte, vol. 19, p. 192, (1941)
Simha, R., "On Relaxation Effects in Amorphous Media", Journal of Applied
Physics, vol 13, pp. 201-207, (1942)
Smith, C. L.,
"A Theory of Transient Creep in Metals", Proceedings of the
Physical Society, vol. 61, pp. 201-205, (1948)
Sykes, R. F. R.,
see
Murgatroyd, J. B., R. F. R. Sykes
Warren, B. E., H. Krutter, and 0. MIorningstar, Journal of. the American
Ceramic Society, vol. 19, p. 202, (1936)
Weber, W., "Ueber die Elasticit~t der Seidenfeden", Pogg. Ann. Physik, (2),
vol. 4, p. 247, (1835)
"Ueber die Elasticitet Fester K8rper", Pogg. Ann. Physik, (2),
vol. 1, p. 24, (1841)
Wiechert, E., "Gezetze der Elastischen Nachwirkung ffir Constante TemperatuD",
Wied. Ann. Physik, vol. 50, pp. 335, 546, (1893)
APPENDIX
AN
ALTERNATIVE
WAY
OF
I
PRESENTING
SURFACE
STRENGTH
RESULTS
Fig. 71 shows an alternative way of presenting the results of a
surface strength test. Each indentation is plotted as a seperate point.
For a given ball diameter D, the ordinate of a point gives a measure
of the load it withstood whereas the abscissa shows the variation in
the position of fracture. The contact circle is plotted as a curve to
show the relation between contact and fracture circles. The slope of a
line connecting the origin to the point of interest in the figure is
proportional to the Hertz stress at the surface.
151
6
-
*0 0
12 i-
10
0*.
P/D
2
/0
e
SO
0
0,*
*
0
*
4p
2 -
0L
0
0.01
0.02
0.03
2
(df/ D)
Fig. 71 Indentation histogram
0.04
0.05-
0.06
APPENDIX
THEORETICAL
LIKELIHOOD
AWAY
FRON
OF
THE
II
FRACTURE
AT
A
DISTANCE
ORIGIN
The indentation stresses outside the contact circle at various
depths close to the surface can be approximately determined by keeping
terms linear in z in equation (2). The change in radial stress as a
function of distance at certain depths is illustrated in Fig. 74.
Fig. 71 shows the locus of zero radial stress as a first approximation.
Considering the radial stresses alone, the function (f (defined as the
difference between the rates of release of elastic energy and the rate
of increase of surface energy for a growing crack) may be plotted for
various depths. If the depth of interest is taken as one micron which would
be of the order of the average Griffith cracks encountered in the Hertz
experiment, Fig. 45 results.
4
i
153
0.2
r
3P
1ra
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
I/a
Fig. 72 Variation
depths
of the radial
below the
tensile stress
surface
at various
n = r/o
0
1.0
1.3
1.2
1.4
1.5
0.1
l= Z/O
0.2
0.3
Fig. 73
Locus of
zero
radial
stress
and its straight
line
approximations
BIOGRAPHY
OF
THE AUTHOR
The author was born in 1930 at Uzunk8pru, Turkey.
He has attended normal secondary schools in Turkey, and in the
year 1948 graduated from the Gazi Lycee in Ankara with top honors in
his class.
He then came to the United States for higher education, and in
the year 1952 graduated from Purdue University "with distinction",
receiving a B. S. degree in Mechanical Engineering.
He continued with his further specialization at the Massachusetts
Institute of Technology wherefrom he received an S. M. degree in the
year 1953
This thesis entitled "Investigations of the Strength and Anelasticity
of Glass" is in partial fulfillment for the requirements of his doctoral
study at the Massachusetts Institute of Technology.
The author is a member of the following honor societies: Phi Eta
Sigma, Pi tau Sigma, Tau Beta Pi, Sigma Xi (associate member).