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:'

This is to certify that the
thesis entitled

STRESS AND TIME EFFECT ON THE CREEP
RATE OF POLYCRYSTALLINE ICE

presented by

SH I OU-SAN KUO

has been accepted towards fulfillment
of the requirements for

PH.D. CIVIL ENGINEERING

degree in

 

 

[Maw

Major professor

Date 9'21'72

 

0-7639

 

 

ABSTRACT

STRESS AND TIME EFFECT ON THE CREEP
RATE OF POLYCRYSTALLINE ICE

BY

Shiou-San Kuo

The purpose of this investigation was to provide
information useful in predicting the time-dependent
deformational behavior of polycrystalline ice.

Creep tests were performed on fine-grained poly-
crystalline ice under uniaxial and multiaxial stress
conditions. Stresses ranged between 0.1586 MNm-2.(23 psi)

and 2.74 MNm-Z

(400 psi). All tests were performed at a
temperature of -4.45°C.

Two forms of ice specimens were used; solid,
cylindrical samples, and thin-walled, hollow, cylindrical
samples.‘ All samples were formed by pouring cold, high
purity water into clean natural snow contained in a pre-
cooled aluminum mold. Ice grain sizes ranged between 1
and 2 millimeters in diameter.

The solid, cylindrical samples were tested

in uniaxial compression only. The thin-walled,

Shiou-San Kuo

hollow, cylindrical samples were tested in uniaxial com-
pression as well as under multiaxial stress conditions.
Creep curves resulting from uniaxial compressive stresses
of 0.686 MNm-2 (100 psi), or greater, exhibited primary,
secondary and tertiary stages of creep. A constant creep
rate was observed in the tertiary stage of creep after

the total strain reached about 10 per cent. This observed
creep behavior is explained in terms of the mobilization
of grain boundary effects, softening of individual crystals
due to accumulated strain on their basal planes, and
recrystallization. By defining a quantity, S, as the
ratio of soft crystals to the total in the polycrystalline
mass, a creep equation was derived which contains three
terms; a primary creep term, a term representing terminal
creep at constant velocity with grain boundary effects
fully develOped, and a third term representing the effect
of a changing S value. This equation appears to fit
experimental uniaxial creep data to strains of at least 25
per cent for the stress range from 0.686 MNm-2 (100 psi)
to 2.74 mm"2 (400 psi).

Computations indicate an apparent activation energy
of 20 Kcal/mole and a flow volume of 2.71 x 104 cubic
angstroms based on the rate process theory.

2

Uniaxial creep tests of stresses below 0.686 MNm-

(100 psi) exhibited only primary and secondary stages of

Shiou-San Kuo

creep. Thus application of the derived creep equation was
inappropriate. However, experimental results agree with
those of Mellor and Testa (1969).

Results from tests on thin-walled, hollow,
cylindrical ice samples confirm the applicability of the
creep equation to multiaxial stress states when stress
and strain are considered in the octahedral sense. It
appears that the resulting principal strain-rates are
parallel and proportional to the corresponding stress

deviators within acceptable experimental limits.

STRESS AND TIME EFFECT ON THE CREEP

RATE OF POLYCRYSTALLINE ICE

BY

Shiou-San Kuo

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Civil Engineering

1972

ACKNOWLEDGMENTS

The writer wishes to express his sincere appreciation
and gratitude to his major professor, Dr. R. R. Goughnour,
Associate Professor of Civil Engineering, for his guidance
and encouragement throughout the preparation and investi-
gation of this thesis. Special thanks are due to Dr. 0. B.
Andersland, Professor of Civil Engineering, for his many
worthwhile suggestions and support during the experimental
work. Thanks are also due to the other members of the
writer's guidance committee; Dr. T. Triffet, Professor of
Applied Mechanics, and Dr. H. F. Bennett, Assistant Pro-
fessor of Geology. To Mr. Leo Szafranski of the Division
of Engineering Research Machine Shep, thanks for his
cooperation in fabrication of experimental equipment.

In addition to those acknowledged above, the writer
wishes to thank the National Science Foundation and the
Division of Engineering Research at Michigan State
University for the financial assistance which made this

research possible.

ii

TABLE OF CONTENTS

ACKNOWLEDGMNTS O O O O C O O O O O O O 0
LIST OF TELES C O O O O O O O O O O O C 0
LIST OF FI GURES O O O O O O O O O O O O O

NOTAT I ON S O O O O O O O O O O O O O O O

' Chapter
I 0 INTRODUCTION 0 O O O O O O O O O 0
II 0 LITERATURE REVIEW 0 O O O I O O O O 0
General Aspects of Creep- . . . . . .
Structure of Ice . . . . . .
Specific Gravity and Specific VOlume . .
The Glide Plane and the Behavior of
Monocrystals. . . . . . . . . .
Compressive Strength of Ice . . . . .
Elastic Behavior of Ice. . . . . .
Rate Process Theory and Its Application .
Theoretical and Experimental Observations
of Mechanical Properties of Ice . . .
III. THEORETICAL ANALYSIS. . . . . . . . .
Uniaxial Creep of Polycrystalline Ice . .
Extension to Multiaxial Creep. . . . .
Plastic Strain Computation. . . . . .
Solid, Cylindrical Samples. . . . .
Thin-Walled, Hollow, Cylindrical

samples 0 O O O O O O O O 0
IV. SAMPLE PREPARATION . . . . . . . . .
Solid, Cylindrical Ice Samples . . . .

Thin-Walled, Hollow, Cylindrical Ice
Samples . . . . . . . . . . .
V. APPARATUS AND EXPERIMENTAL PROCEDURES . . .

Solid, Cylindrical Ice Samples . . . .
Thin-Walled, Hollow, Cylindrical Ice
Samples . . . . . . . . . . .

iii

Page

ii

vi

ix

10
12

13
15
16
19
27
43
43
60
64
64
66
69
69
71
74
75

81

Chapter Page
VI. RESULTS AND DISCUSSION . . . . . . . . 89

Solid, Cylindrical Ice Samples . . . . 89
High Stress Level. . . . . . . . 89

Low Stress Level . . . . . . . 103
Thin-Walled, Hollow, Cylindrical Ice
Samples . . . . . . . . . . . 105
Constant Axial Stress Creep Test. . . 105
Constant Tensile Creep Test . . . . 106
Constant Biaxial Stress Creep Test . . 107
VII. CONCLUSIONS. . . . . . . . . . . . 150

BIBLIOGRAPHY O O O O O O O O O O C O O O 153

APPENDIX. 0 O O O O O O O O O O O O O O 159

iv

LIST OF TABLES

Table Page

1. Parameters of Equation (3-25) Evaluated for
Best Fit to Series B Tests, T = -4.45°C. . 143

2. Parameters of Equation (3-25) Evaluated for
Best Fit to Series B Tests, with C and

C Values Fixed . . . . . . .1 . . . 144

3. Results of Tests to Determine Value of
Recrystallization Constant K '. A11 _2
Tests at -4.45°C and Stress of 1.71MNm
(250 psi). . . . . . . . . . . . 145

4. Values of K ', 85’ K ', C ', C ', C3', and

KZ'So for a Range of S resses . . . . . 146
5. Value of S as a Function of Stress and Time . 147
6. Parameters of Equation (3-25) Evaluated for

Best Fit to Sample 1H. . . . . . . . 148

7. Octahedral Parameters Evaluated for Best_
Fit to Sample 3H Compared with Series B
Results . . . . . . . . . . . . 149

Figure

2-1.

2-4.

2-5.
2-6.
3-1 0

3-2.

3-3.
4-1.

5-6.

LIST OF FIGURES

Schematic RepreSentation of a Typical Creep
curve C O O O O O I O C O O O 0

Structure of Ice Lattice (After Pounder, 1961)

Stress, Shear Stress and Glide Plane.

Temperature Dependence of Young's Modulus for
Polycrystalline Ice (After Gold, 1958) . .

Schematic of Units of Flow . . . . . . .
Schematic Representation of Energy Barrier. .
Schematic Diagram of Creep Curve Characterized

as Two Parts. . . . . . . . . . .

Stress on Ice Crystals in Polycrystalline
Mass . . . . . . . . . . . . .

Schematic Diagram of Two Phase Loading . . .

Disassembled Mold for Forming Thin-Walled,
Hollow, Cylindrical Ice Samples . . . .

Triaxial Cell, Test Sample, and Mold for
Solid, Cylindrical Ice Samples. . . . .

Schematic Diagram of Axial Loading System . .

Triaxial Cell in Coolant, Text Machine,
Cooling System and Sanborn Recorder . . .

Deformed Solid, Cylindrical Ice Sample
Compared with an Undeformed Sample . . .

Schematic Diagram of Triaxial Cell Modified
for Testing of Thin-Walled, Hollow,
Cylindrical Ice Samples . . . . . . .

Nitrogen Cylinder and Testing Apparatus. . .

vi

Page

11

13

18
20
22

45

45

55

71

75

76

78

82

83
86

Figure Page

5-7. Deformed Thin-Walled, Hollow, Cylindrical
Sample Compared with an Undeformed Sample . 88

6-1. Creep Curves for Five Series A Samples . . . 111

6-2. Axial Strain-Rate Versus Axial Strain,
Sample 4B. . . . . . . . . . . . 112

6-3. Creep Curve for Sample 6A
(a) Showing Three Loading Cycles
(b) Loading Cycles Superimposed . . . . 113

6-4. Creep Curve for Several Series B Samples
(a) Sample 1B, (b) Sample ZB,
(c) Sample 3B, (d) Samples 4B and 5B,
(e) Samples 6B, 7B, BB and 9B . . . . . 114

6-5. Creep Curve for Sample 8B Showing Correlation
With Equation (3-25) 0 o o o o o a o 119

6-6. Best Fit Values of C as a Function of Stress. 120

1
6-7. Best Fit Values of C from Equation (3-25)
Versus Axial Stress . . . . . . . . 121

6-8. Best Fit Values of n from Equation (3-25)
Versus Axial Stress . . . . . . . . 122

6-9. Best Fit Values of m from Equation (3-25)
Versus Axial Stress . . . . . . . . 123

6-10. Best Fit Values of C from Equation (3-25)
Versus Axial Stre s . . . . . . . . 124

6-11. BeSt Fit Values of C from Equation (3-25)
-Versus Axial Stre s . . . . . . . . 125

6-12. Axial Strain Necessary for Softening Term
to Reach Stated Percent of its Original
Value as a Function of Stress . . . . 126

6-13. Creep Curves for Sample 10B Showing Effect of
Abruptly Increasing Load
(a) First Loading Phase
(b) Second Loading Phase . . . . . . 127

6-14. Creep Curves for Sample 118 Showing Effect of
Abruptly Decreasing Load
(a) First Loading Phase
(b) Second Loading Phase. . . . . . . 128

vii

6-27 0

6-28.

Creep Curves for Sample 12B, 13B, 14B and 153
in Second Phase Loading . . . . . . .

Values of K2' Versus Axial Stress. . . . .

Plot of Equilibrium Values of S Versus Axial
stress. 0 O O O O O O O O O O O

Computed KZ'SS, C2' and K2' So Versus Axial
stress. 0 O O O O O O O O O I O

Creep Curve for Polycrystalline Sample 16B. .

Axial Strain-Rate Versus Axial Stress
(After Mellor and Testa, 1969). . . . .

Creep Curve for Sample 1H . . . . . . .
Creep Curve for Sample 2H (Tensile Creep Test)
Creep Curve for Sample 3H . . . . . . .

Octahedral Shear Strain Versus Time for
Sample 3H. . . . . . . . . . . .

Plot of noct Versus Octahedral Shear Stress

for Samples 1B through 9B Compared with
Sample 3H. . . . . . . . . .. . .

Plot of moct Versus Octahedral Shear Stress

for Samples 1B through 93 Compared with
sample 3H. 0 O O O O O O I I O 0

Plot of C oct. Versus Octahedral Shear Stress
for Samples lB through 9B Compared with
sample 3H. 0 O O O O O O O O O 0

Schematic Representation of Vectors of Stress
Deviator and Strain-Rate for Sample 3H . .

viii

Page

129
130

131

132

133

134
135
136
137

138

139

140

141

142

O 3’ 3’0

AF

GS

AH

sun 0 ”U

NOTATIONS
angstrom units = 10"8 cm
effective area represented by one flow unit
elastic deformation
initial inner diameter of hollow cylinder
inner diameter change in hollow cylinder»

Young's Modulus, energy provided by applied
shear stress

free activation energy

system error

grain size

heat of activation

Planck's constant = 6.624 x 10.27 erg. sec.‘
Boltzmann's constant = 1.3805 x 10-16 erg/°C
specific rate of the process

net relative distance moved by one flow unit in one
transition

Avogadro's number = 6.02 x 1023/g mole; number of
dislocation

hydrostatic pressure
free activation energy
stress concentration factor

universal gas constant = 1.987 cal/mole/°C

ix

85
AS

St

oct.

Yoct.

Yoct.

radius of thin-walled, hollow, cylindrical sample
ice density

a structure term

equilibrium value of S

entrOpy of activation

tensile hoop stress

temperature in °C or absolute temperature

time; wall thickness of thin-walled, hollow,
cylindrical sample

initial volume

current volume

true axial strain

initial elastic strain

rupture strain

first principal strain, primary creep strain

second principal strain, creep strain with grain
boundary effects fully developed

third principal strain
axial strain rate
conventional strain
shear stress

octahedral shear stress
shear strain

shear strain rate
octahedral shear strain

octahedral shear strain rate

distance between equilibrium positions of flow units

distance between flow units normal to direction of
flow

normal frequency
coefficient of viscosity

angle between projection of direction onto the
glide plane and nearest direction of glide

angle between 0 direction and glide plane
axial stress

first principal stress

second principal stress

third principal stress

stress deviator

xi

CHAPTER I

INTRODUCTION

The most highly develOped branch of ice study, and
at the same time the only branch that has long been an
independent scientific discipline is the study of glaciers,
which is called glaciology by West European tradition.

But in Latin-Greek the word "glaciology" means literally

"the study of ice,‘ and consequently, the study of ice in
general, not merely glaciers. Ice structure, as an object
of study in structural glaciology, should be understood in
the broad sense of the word, beginning with the so-called
fine structure of crystals, i.e., the structure of the Space
lattice of ice, including structure and texture in the
petrographic sense, and going as far as tectonic structures.
Pure ice as a crystalline substance is an abstraction. The
properties of natural ice bodies are due to the presence of
other substances in them, and to the structure of the ice
as an aggregate of crystals of a particular form, dimension
and mutual orientation. These features in turn are a
function of the formation process of the given ice body.

Therefore, the physical study of the fine structure and of

the constants of the ice mineral is only a beginning, the

main requirement being the structural genetic study of
ice as a rock.

The recent increased activity in the Arctic and
Antarctic regions, where ice is used as a construction
material, has caused an interest in the mechanical behavior
of ice. This interest had previously been linked to the
subject of glaciers, and it was long thought the flow of
glaciers was associated with regelation. In order to
understand the flow of glaciers and ice bodies it is
necessary to know at what rate ice deforms under various
stresses. Thus considerable work has been done on the
deformation of both single crystal and polycrystalline
ice, including observation of glacier flow as well as
laboratory experiments. The failure of conventional
cencepts to describe the time dependent deformation of
single crystal and polycrystalline ice has forced many
investigators to examine the microsc0pic aspects of creep
behavior. Deformation of single crystal ice seems to be
controlled by the movement of dislocations on the basal
plane. However, the flow of polycrystalline ice cannot
be explained so simply. The purpose of this study is to
contribute knowledge toward the understanding of the
mechanical behavior of polycrystalline ice subjected to an
applied stress.

Several creep theories have been proposed by many

investigators on the basis of experimental findings. The

most generally accepted concept is that creep behavior is
strongly temperature dependent and is governed by thermally
activated processes. Following Kanter's (1938) proposal,
theoretical work on creep was influenced by the rate-
process theory (Glasstone gt_al., 1941) which was employed
to define steady-state creep in metals, and was latter
applied to describe the steady-state creep of a large
number of materials including ice (Glen, 1954; Steinemann,
1954; Butkovich and Landauer, 1960; Dillon and Andersland,
1966; Goughnour and Andersland, 1968, etc.). Goughnour
and Andersland (1968) further suggest that the creep
behavior of polycrystalline ice in uniaxial compression

is dependent in part on absorbed energy.

When a relatively high stress is applied to poly-
crystalline ice, after an immediate elastic deformation,
the creep curve exhibits a primary or transient creep
stage, a secondary or steady-state creep stage, and a
tertiary or accelerating creep stage. Typica1.creep
curves for this type of performance are shown on Figure 6-4.
However, this entire sequence is not always observed
during creep of polycrystalline ice. Whether all creep
stages are exhibited in any one test depends on temperature,
stress, and the duration of the test. A typical creep
curve for lower stress (0.1586 MNm-2 = 23 psi) is shown on
Figure 6-19, which exhibits only primary and secondary

stages of creep. These phenomenological aspects of creep

for polycrystalline ice at high and low stress are dis-
cussed in the chapters on literature review and experimental
results.

All the samples for this study were prepared by
freezing a mixture of natural snow and precooled pure water.
These techniques are described more fully in later chapters.
Experimental data obtained from this study indicate that
the creep rate of polycrystalline ice indeed depends on the
stress, strain, history of loading and temperature. The
results of uniaxial compressive creep tests under a certain
range of high stresses suggest that the creep rate may
approach some constant value at large strains when grain
boundary effects are fully develOped and the structure of
the ice approaches some equilibrium value. This equilibrium
structure value is believed to result from a balance between
the creation of soft crystals, created by deformation along
their basal planes, and recrystallization. By using this
concept along with equation (2-34) developed by Goughnour
and Andersland (1968), a creep equation for higher stresses
on polycrystalline ice is derived. Data from constant
"axial stress creep tests were used to evaluate the param-
eters in this equation which appears to account for the
creep behavior of polycrystalline ice throughout the primary,
secondary and tertiary stages of creep for the range of

stresses and temperature (-4.45°C) studied.

Uniaxial compressive creep tests at low stresses
on solid, cylindrical polycrystalline ice samples were also
performed. The constant strain-rate portion of these
tests agree very well with the results presented by Mellor
and Testa (1969). However, it is believed that the creep
equation derived from this study is applicable to low
stress tests also, if the duration of creep is long enough.
Thin-walled, hollow, cylindrical polycrystalline
ice samples were also tested under biaxial stress condi-
tions. These results were used to evaluate the validity
of the assumption (Nye, 1953), that for polycrystalline
ice the principal strain rates are parallel and proportional
to the corresponding principal stress deviators.
The applicability of the creep equation derived in

this study to multi-axial stress states is also verified.

CHAPTER II

LITERATURE REVIEW

Experimental investigations into the creep behavior
of ice have followed two main lines of inquiry; behavior
of ice single-crystals, and behavior of polycrystalline
ice. Several theories have been presented to describe the
creep behavior of each. However, only limited success has
been attained in correlating the behavior of single ice
crystals to that of polycrystalline ice.

Chapter II will present a brief review of current
concepts of time-dependent deformational behavior of
materials in general, and a more specific discussion
regarding the creep behavior of both single and poly-

crystalline ice.

General Aspects of Creep

 

A specimen undergoing continuous deformation under
a constant load or stress is said to creep. This phenome-
non is observed both in metals and nonmetals. In general,
creep refers to strain that is time-dependent and may
include elastic, viscous and plastic deformations. From
a macroscopic View point, creep deformation is often
treated as plastic deformation, and elastic strain and

viscous flow are neglected.

 

Creep of metals as well as nonmetals can be demon-
strated directly by a creep curve which represents graphi-
cally the function between creep strain and time. Its
slope at a specific time is called creep-rate or strain-

rate. A typical creep curve is shown in Figure 2-1.

Tertiary

I

I

   
    

Secondary

 

_._IL

   

Strain

 

 

 

rr b-—— —-————o — —

Time

Figure 2-1.--Schematic Representation of a
Typical Creep Curve.

The shape of the creep curve as shown in Figure 2-1
has led to a subdivision of the creep curve into four

parts:

l. Instantaneous strain, 80, which is obtained
immediately upon loading and exhibits character-
istics cf plastic deformation, but includes
elastic deformation.

2. Primary creep which is between 80 and £1. The
creep rate decreases continually.

3. Secondary or steady-state creep which is
between 81 and £2. The creep rate remains
constant, indicating a nearly steady-state
condition.

4. Tertiary creep which is beyond 6 The creep

2.
rate increases continually, leading eventually
to rupture at strain Er and time tr' This
behavior is thought to be a result of inter-
crystalline movement as opposed to the intra-
crystalline movement of primary and secondary
creep. Tertiary creep can occur at any
temperature providing the stress level is
relatively high.

Whether all creep stages are observed in any one
test depends on stress, temperature and the duration of the
test. Different materials will show similar creep curves I
under proper adjustment of stress and temperature. Creep

curves for metals, crystalline nonmetals and noncrystalline

materials are all similar. Considering the differences in

atomic structure of the materials, it must be evident

that the details of the creep curve itself cannot identify
the micro-mechanisms responsible for creep. However, the
similarity of the creep curves for these materials does
indicate that a similar sequence of rate determining
mechanisms is followed in all cases.

Experimental evidence conclusively shows that creep
flow is thermally activated; local thermal agitation pro-
vides added energy, beyond that provided mechanically, to
overcome barriers to creep deformation. Then the applied
stress aids in overcoming these barriers but serves also
to give direction to the resultant flow. A generalized
function for the creep rate, 6, based on the theory of

thermodynamic fluctuation can be given by

-AFi(T,S)

é = 1 z1 (v,T,S)oi (T,S)exp [-——§§————] (2-1)

The function, 21' may depend on the temperature,T, and
contains the normal frequency, v, for the flow unit, the
entropy change and a structure term S. The latter involves
such microstructural features as grain size, precipitation
and dispersion of one or more phases, density, distribution,
and length of dislocation. The stress function oi may
depend on temperature and structure because creep mechanisms

are temperature- and structure-dependent. AFi governs the

creep rate and is temperature and structure-dependent.

10

R is the universal gas constant. From equation (2-1) it
is clear that creep is a Complex phenomenon, and that the
evaluation of this function requires extensive experimental

and theoretical work (Frank Garofalo, 1965).

Structure of Ice

 

The prOperties of ice, as is the case with all
crystalline materials, are determined not only by its
chemical composition, but also by the geometry of its
space lattice and the nature of the bond that keeps the
structural units of its space lattice in equilibrium.

The problem of the fine structure of ordinary ice
has not yet been solved completely. X-ray analysis can
establish only the position of the oxygen atoms in the
space lattice of the ice. The hydrogen atoms do not
scatter x-rays sufficiently, and at present their position
is judged primarily from indirect data. The position of
hydrogen atoms changes continuously at temperatures above
approximately -70°C, so that only the average.structure
determined by the position of the oxygen atoms can be
recorded.

The unit cell of this average structure consists
of four oxygen atoms. These oxygen atoms are formed in a
tetrahedral pattern, with each oxygen atom being surrounded
by four approximately equally spaced oxygen atoms at the

vertices of the tetrahedron. Each pair of oxygen atoms

11

is linked by a hydrogen bond (Pounder, 1965). The unit

cell of ice is depicted on Figure 2-2.

0'" OI"

Olll

 

0.923 A

 

2.760 A

 

 

 

 

 

Figure 2-2.--Structure of Ice Lattice. (After
Pounder, 1961).

If the tetrahedron were perfect, that is if all
bounds were of equal length, and all the 0'00" and O"OO"
angles were equal, these interior angles of the tetra-
hedron would be 109° 28'. Solid lines shown on Figure 2—2
represent hydrogen bonds and the outline of the tetra-
hedron surrounding the oxygen atom marked 0 is shown by
dashed lines. The three atoms marked 0" form an equi-
lateral triangle in a plane which is called the basal

plane of the lattice. The 0'0 bond gives the direction

12

I
known as the’C-axis, which is perpendicular to the basal

plane. Considering now the four bonds of the atom 0', one
is O'C, and the other three are O'O"'. These three atoms
must also lie in a plane perpendicular to 0'0. Thus the
tetrahedron about 0' is inverted, with its base parallel
to that of the tetrahedron about 0.

Figure 2-2 shows the complete bonding system for
the O and 0' atoms, but each of the three 0" and 0"' atoms
must have other three additional bonds. In this way the
lattice is extended in all three directions. This orderly
arrangement of atoms is disrupted at grain boundaries.

Masses of polycrystalline ice are aggregates of
single crystalline grains. In quickly frozen masses these
grains may range from only a fraction of a millimeter to
several centimeters in size. The boundaries between
neighboring crystals in polycrystalline ice are usually

irregular and often depend on the mode of freezing.

Specific Gravity and Specific Volume
Ice has an "open," not very compact structure.
Therefore it is lighter than water. The specific gravity
of pure ice at 0°C and 1 atm pressure is 0.9168 gram/cm3,
and its specific volume is 1.0908 cm3/gram (Bridgman,
1912; Lonsdale, 1950), while the specific gravity and
specific volume of water are 0.999863 gram/cm3 and 1.000132

cm3/gram, respectively. Any departure from the above

13

value for ice indicates the presence of impurities, or a

different temperature.

The Glide Plane and the Behavior
of'Monocrystals

 

 

The basal plane of an ice crystal is its well
defined glide plane. The stress, 0, at the elastic limit
differs from the critical tangential stress I as a
function of the angle ¢ between the c direction and the
glide plane, and also as a function of the angle w between
the projection of the c direction onto the glide plane and

the nearest direction of glide:(see Figure 2-3)

I = o(sin ¢ cos 0) (2-2)

 

 

L\‘<::z"glide plane

Figure 2-3.--Stress, Shear Stress and Glide Plane.

In ice the magnitude of the critical normal stress
p necessary to rupture the crystal along the basal plane

is incomparably greater than the critical tangential stress

14

at which gliding starts. If the critical tangential
stress I is reached, the ice will deform plastically,
without disruption of continuity; if, however, the criti-
cal normal stress p is reached at the same time or subse-
quently during the plastic deformation process, the ice
will act as a brittle body and split. A force causing
shear may act in three main, mutually perpendicular
directions (McConnel, 1890);

1., The shear force coincides with the basal

‘plane, only translation occurs, the deforma-

tion is plastic;

Shear force

IIIII lllll Main axis

2. The shear force acts in the direction of main
axis perpendicular to the basal plane. The
elementary plates of the crystal bend and break
after the critical stress is exceeded. The

deformation is elastic-plastic;

15

Shear force

"lI-IIIIIII"
.
v

Main axis

3. The force acts in the basal plane, but the
shear force is perpendicular to it. The
elementary plates that are rigid in the longi-
tudinal direction allowonly a negligible
elastic deformation, which changes to disrup-
tion in the case of a significant increase of
stress. Generally there is no plastic deforma-

tion in this case.

Compressive Strength of Ice

 

The compressive strength of ice has been extensively
investigated. An excellent summary of the results prior to
1951 may be found in the companion volume of U.S. Army
Snow,Ice and Permafrost Research Establishment Report
Number 8 (Volume II, Appendix B, Table 12). According to
the report, Vitman and Shandrikov found a general increase
in the strength of ice as the temperature is decreased. A
sample having a number of small crystals has a greater
strength than a sample having one or more crystals.

Butkovich (1954) performed a series of tests on

cylindrical ice specimens to determine the ultimate

16

strength of ice. He used lake ice, natural snow ice, and
commercial ice. His results showed conclusively that
temperature has a pronounced effect on the strength of
ice. His results produced a strength variation of as much
as 140 psi from the mean strength for ten samples. An
important conclusion is that the strength is indeed
dependent on how the load is applied to the ice itself.

Leonards and Andersland (1960) found that the
strength of ice increases approximately 15 psi per degree
centigrade decrease in their unconfined compression tests
using identical, polycrystalline cylindrical samples.
Their results showed good agreement with those of

Butkovich (1954).

Elastic Behavior of Ice

 

The determination of Young's modulus for ice has
been attempted by two methods, static and dynamic.

Dorsey (1940) questioned whether significantly useful
values can be obtained by means of the static method.
There are several factors which could affect the values
of Young's modulus: temperature, loading rate, grain
size, orientation, and density.

Many of the early static determinations of Young's
modulus for ice were done at temperatures only slightly
below 0°C, and those tests which were performed at tempera-
tures below 0°C were done with little concern for the

control of temperature. The results from U.S. Army Snow,

17

Ice and Permafrost Research Establishment (1952) show
that as temperature decreases, the value of Young's
modulus increases slowly. Their linear relationship
gives an E value of 1.25 x 106 psi at 32°F and 1.52 x 106
'psi at -34°F. These values were obtained from both sonic
and static test methods. Nakaya (1959) observed a hysteresis
phenomenon in the relationship between Young's modulus and
temperature.

In most polycrystalline aggregates the likelihood
of large dependence on orientation is small. However,
Young's modulus depends on the preferred orientation of
the grains (Van Vlack, 1959) and its value from specimenS‘
prepared with random orientations will represent an approxi-
mate average value. Gold (1958) performed experiments
to determine the elastic modulus of ice by a static
method. He took care to control both temperature and
grain size. Figure 2-4 shows his results plotted as
Young's modulus versus temperature. It is seen from
Figure 2-4 that as the grain size increases, the modulus
increases.

Dynamic measurements of Young's modulus were also
cede With little attention given to control of the experi-
mental variables. However, at the higher "loading rates"
used in the dynamic method, the temperature dependence
‘was a full order of magnitude less than that observed for

static meaSurements.f This result is shown on Figure 2-4

1.4x106
1.3
'5:
,3 1.2
m
3
, 1.1
8
5‘ 1.0
P
m
5 0.9
O
>0
0.8

18

 

 

 

 

 

 

10.0x1010
L E sonic = 9.0 x 1010 to 10 x 1010 dyn/cm2
- Ex =- 8.34 x 101'0 dyn/cm27 7 9'0
l
l
——A——oa—%1 44—33——
‘9 8 as 4 2 “s 8 ”a
- .0 O
. a
8'
I- q 7.0 g
'5‘
£19 3
I E:
13.1 12 .. 6.0
-2 10 2 o s .9
_ Ex I (5.69 - 6.58 x 10 T)x10 dyn/cm. a.
. 5
1 J n n L 1 J 5.0 g
-40 -35 -30 -25 -20 -15 -10 -5 0
Temperature °C
LEGEND

 

[3 Single Crystal

.Fine Grain (diam. < 0.50 cm)

Medium Grain (0.50 < diam. < 1.50 cm)

D> D' C)

Large Grain (diam. > 1.50 cm)

0 Average Grain diameter not observed

Figure 2-4.--Temperature Dependence 6f Young's
Modulus for Polycrystalline Ice (After Gold, 1958).

19

also. The dynamic values are seen to be very consistent.
Gold (1958) stated that doubling and halving the loading
rate did not appreciably change the Young's modulus values.
Halvorsen (1959) found that his values of elastic modulus
varied between 1.735 x 105 psi and 1.94 x 105 psi as the
loading rate varied.

The density of ice has a pronounced effect on the
value of Young's modulus (Nakaya, 1959; Halvorsen, 1959).
As the density decreases, a corresponding decrease in the
value of Young's modulus occurs. Nakaya's results show
that for densities between 0.915 and 0.903 Young's modulus

6

varies from 1.33 x 10 psi to 1.09 x 106 psi respectively.

His results agree well with those of previous investigators.

Rate Process Theory and Its Application

 

The reaction-rate theory (Glasstone, Laidler, and
Eyring, 1941) which was employed earlier by Eyring (1936)
to predict viscosity, plasticity and diffusion in viscous
fluids has greatly influenced the theoretical work on
creep. The reaction-rate theory involves the concept of
activated complexes and is based on statistical mechanics.
The formation of activated complexes requires energy. A
complex is regarded as being situated at the top of a free
energy barrier lying between the initial and the final
states. It is assumed that the "unit of shear process"
involves the motion of "units of flow" which are regarded
as elementary structures within a solid (atoms, molecules

or aggregates of molecules). The motion of these flow

20

units passing from one equilibrium position to another
constitutes the shear process, but can only be accom-
plished by overcoming energy barriers. This barrier can
be overcome by activating the flow units." A schematic
representation of units of flow is shown on Figure 2-5,
where T is the shear stress, 1 is the distance between

equilibrium positions of flow units, and Al is the distance

 

l

 

{er
tom
1.

 

 

 

 

 

Figure 2-5.--Schematic of Units of Flow.

between flow units normal to the direction of flow. If
K' is the net number of transitions per second made in

the direction of shear, then the shear rate dy/dt becomes

21%
I

K'. (2-3)

>a
rely

An expression for the specific reaction rate
equation of molecular processes based on the rate process

theory is

21

K' = %? e-AF/RT (2-4)
where K' is the specific rate of the process, k is
Boltzmann's constant equal to 1.3805 x 10"16 erg/°C/mole,

T is the absolute temperature, h is Planck's constant
equal to 6.624 x 10.2 erg. sec., AF is the free activation
energy, and R is the universal gas constant equal to

1.987 cal/degree mole.

In the absence of an applied shear stress to the
material, a state of equilibrium exists and the number of
transitions in one direction is equal to those in the
opposite direction. If, however, a directed shear stress
is applied to the material, the energy barrier becomes
distorted. Kauzmann (1941) assumed that the activation
energy for motion of units of flow in the direction of
shear is lowered by the applied shear stress and the change
in activation energy is proportional to the stress. In
the direction opposite to the direction of flow the
Tactivation energy is raised by an equal amount. Figure
2-6 depicts the energy barrier schematically. In Figure
2-6 the solid line represents the energy barrier in the
absence of applied shear stress, and the dashed line shows
the distortion of the barriers when shear stress is
applied. The energy, E, provided by the applied shear

Stress is given by

Potential Energy

 

22

Shear Stress T

 

 

 

 

 

 

' —* I
' I
I ' I I
With applied l
shear I l Without applied
I I, shear
I I
I I I I
\ I ’1’ AF \ I
TqA£ —I I I I
2 \_1| \ I
2/2 \\ 14:2
I ~ ,’
£

 

 

 

 

Direction of Strain

Figure 2-6.--Schematic Representation of_Energy
Barrier.

E = rqu (2-5)

where A is the projected area of the unit of flow on the
shear plane, 1 is the distance the shear stress acts in
surmounting the potential barrier, and q is a stress con-
centration factor. This is the energy required to
displace a flow unit from its initial state to its final

state.

23

As a result of the applied stress on the unit of

flow, the barrier height in the forward direction is

(AF - l%?£) and in the backward direction is (AF + 1%?£).

The net specific rate in the forward direction is then

kT AF IqAR kT TgAR)

- —- AF
I _ I __ __ _ __ __ .-
K K 11 9X9 I RT + 2RT) 11 EXP I RT' 2RT

2kT

_ A2. . TgAR
_H— exp ( RT) Sin h

ZRT'

(2-6)

By substituting this into equation (2-3) the shear strain-V

rate becomes

-_A___2kT -92: .
- A1 h exp ( RT) Sin h

TgA2

ZRT‘ (2'7)

.< .
ll
9:?

Herrin and Jones (1963) indicated that A and Al are only
approximations of the order of the size of a flow unit,
and it may be assumed that A equals Al. Hence equation

(2-7) becomes

2kT

' . g l
y = -3— exp (-%%) Sin h T A

2RT' (2'8)

Since k (Boltzmann's constant) equals R/N, where N is
Avogadro's number (6.02 x 1023/9 mole), changing the units

of equation (2-8), one obtains

°_ 2kT _ £3. . TgA£ _
7“ h exP I RT) Sin h 2kT' (2 9)

24

From thermodynamics, the free energy of activa-

tion AF can be expressed as a function of temperature,

AF = AH - TAS ' (2-10)

where AH is the heat of activation, and AS is the entrOpy
of activation. Substituting equation (2-10) into

equation (2-9) gives

_ 32_ '-AH + TAS . TgAR _
y — 2 11 exp ( RT ) Sln h 2kT (2 ll)

 

or

I = 2%? exp (%§) exp (- %¥) sin h l%%%u (2-12)

This is the general creep equation derived by using the

rate process theory, and is based on a single molecular
creep mechanism. However, this need not always be true
since in a given material different mechanisms may be
operating simultaneously, each having its own characteristic
values of AH, AS, q, A, and 2 (Kauzmann, 1941). Thus

.equation (2-12) may be written as

AS. AH. (qu).I
'_ 152 _.1_ -.__1.- ___1_ -
y -i 2 11 exp ( R ) exp ( RT) Sln h 2kT (2 13)

where the sum is over all possible mechanisms. Conrad
(1961) pointed out that usually only one of the mechanisms

(i.e., only one of the terms in equation (2-13)) will

25

account for the greater part of the observed flow rate,
but it is possible that a single term will not give a
major part of the shear rate under all conditions of stress
and temperature. For example, one mechanism may account
for most of the creep under low stresses, while under
large stresses a different mechanism may dominate. Each
mechanism contributes something to the total observed rate
under all conditions.

When the applied shear stress is small compared
to the thermal energy,

Tqu TgA£

31“ h 2kT ~ 2kT.

Equation (2-9)becomes

° _ 2kT _ A§_ Tqu
Y ‘ h exP I RT) 2kT

or

.. Tel-ftfiv eXp (_ %). (2'14)

.4
I

That is the shear strain rate is proportional to the shear
stress which constitutes Newtonian flow. The coefficient

of viscosity can be expressed as

_ h.. A}: ..
n — qu exP (RT). (2 15)

26

When the energy supplied by the applied shear stress

is much greater than thermal energy, then

. TgAl z 1_ TgAl.
31“ h 2Kt 2 ex? 2kT'

Equation (2-9) will become
° kT AF 2
'Y = -h— exp (" .1371?) EXP ("l'_2?<_%_) . (2‘16)

Equation (2-16) is a very useful form in the study of the
creep proCess.

Equation (2-12) can also be written to the form

y = A Sin h B (2-17)
.. M 9.5. _ All
where A — h exp (1!) exp ( RT), and
_ 2kT
B — qAR'o

Born and his co-workers (1954) have shown that for poly-
crystalline aluminum, B is independent of temperature and
stress and appears to be independent of the structure of
metal. A is dependent on temperature, structure and stress
(Conrad, 1961). Dorn and his co-worker (1954) found the

creep rate equation as

-AH/RT Bo
e e

m.

= 5p (2-18)

27

Where the stress function eBC is the effect of stress on
the deformation process itself and not on the structure,

Sp is the effect of structure on the creep rate.

Theoretical and Experimental
Observations of_MechanicaI
Properties of Ice

 

Investigators have studied the flow of ice, as
influenced by stress and temperature by observing the
deformation of ice single crystals and polycrystalline
ice. In 1887 Main performed deformation experiments on
ice samples. He showed that polycrystalline ice would
flow under conditions that would preclude regelation,
but his control of temperature and other variables pre—
cluded any quantitative conclusions. McConnel and Kidd
(1888) followed this work to confirm Main's investigations.
In addition, they found a marked temperature dependence of
the flow rate of polycrystalline ice. They did work also
on single ice crystals, and found that when the C-axis
is perpendicular to the stress, the crystal yields only
slightly. However, they felt that ice crystals were not
plastic and attributed to the flow of polycrystalline ice
to "some action at the interface of the crystal."

McConnel (1891) continued the work on single crystals and
found that ice crystals are plastic and slide on the basal
plane. He likened ice to an "infinite number of thin

sheets of paper, normal to the optic axis." If ice

28

deforms more or less perpendicularly to the main axis, the
slight elastic stress relaxes intensively, while if the
deformations are parallel to the main axis, the relaxation
is weaker. Hawkes (1930) found that deformation in ice is
due to slip along (1) basal plane, (2) grain boundaries,
‘and (3) fracture surfaces. I
In recent years, an increasing number of researchers
have investigated the flow of ice. Glen and Perutz (1954),
and Steinemann (1954) have shown that ice deforms plastically
only by translational gliding on the basal plane (0001),
which confirms the original observations of McConnel (1891).
Steinemann (1954) found that ice single crystals subjected
to shear stress parallel to the basal plane undergo a
transition in creep behavior at strains between 10 to 20
per cent. This transition sometimes occurs abruptly and
at other times is gradual, but it always results ulti-
mately in a drastically increased yet uniform creep rate.
The relation between stress, strain and time for
ice single crystals has been investigated by several experi-
menters. Griggs and Coles (1954) performed creep tests on
single crystals for which the basal plane was oriented at
45° to the compression axis. They found that their creep

data could be best fitted to an equatiOn of the form
a = a[(o - sz) t]2 (2-19)

'where e is the compreSsive strain, 0 is the stress, T is

the temperature below the melting point and t is the time.

29

If e is expressed in per cent, a in Kg/cm.2, T in °C and t
in hours, then a equals 0.62 and b equals 0.02.'

Steinemann (1954) carried out his tests on ice
single crystals for more than 20 hours and found that his
creep curves appeared to settle down to a steady rate of
flow rather than continue to increase parabolically, and he
has related steady-state rate by plotting shear strain
rate Y against shear stress T logarithmically. He found
a relationship of the form § = KTn, where K and n are
constants. Butkovich and Landauer (1959) also found that
the flow law, § = KTn, provided a best fit to data from
shear tests of single crystals in "easy glide." Steinemann
found a value of n at low strains ranging from 2.3 to 3.9,
and at higher strains of 1.5. Griggs and Coles (1954)
indicated a stress-dependence of the flow rate corresponding
to an n value of 2.0. Butkovich and Landauer (1959) found
n average of 1.7. Kamb (1961) performed torsion tests of
hollow single crystal ice cylinders and deformed the
specimen to several hundred per cent at a temperature
-2.9°C. He found a value of n equal to 2.5. Jellinek and
Brill (1956) applied a different approach. They performed
tension creep tests on single crystals with their optic
axes oriented at 45° to the sample axes. They found that
some of the samples followed a sat2 law, whereas others
showed a linear relation between strain and time. They

assumed first that

30

e = KlN (2-20)

where N is the number of dislocation in the specimen at

time t, and then

u = K

or

N = KZt + NO (2-21)

where No was the number of dislocation present at the
start of the creep test. From equations (2-21) and (2-20)

one obtains

a = Klet + KlNO. (2-22)

Integrating equation (2-22) gives

H

_ _ ' 2 _
e — 2 Klet + KlNot + so (2 23)

where so is the instantaneous elastic strain. If the

Second term is negligible, a parabola is obtained. However,
if No is quite large, then the linear term might dominate
and a linear relationship results. As the number of dis-
locations increases with time, they will mutually annihilate
each other and finally reach steady-state creep. This was

also observed by Steinemann (1954) as mentioned previously.

31

Higashi and Sakai (1952) performed experiments on.
the movement of low-angle tilt boundaries in ice and found
an activation energy between 12 and 17 Kcal/mole. Itagaki
(1966) found that the activation energy for self-diffusion
in ice varied from 12.6 Kcal/mole to 15.4 Kcal/mole.

In polycrystalline ice, where the orientations of
individual grains are random, continuity demands that
mechanisms other than basal slip operate to permit the
grains to conform to an arbitrary change of shape. Glen
(1952) reported on the results of compressive creep tests
on blocks of polycrystalline ice at -l.5°C. His results
showed that polycrystalline ice does not possess a con-
stant coefficient of viscosity, but obeys a relationship
between compressive stress, a, and strain rate, é, of the

form

é = kcn (2-24)

where n is a conStant equal to about 4. Later in 1955,
Glen continued his experiments on the creep of poly-
crystalline ice under compressive stress conditions and
found that the relationship between the minimum observed
creep rate, s, the applied stress, 0, and the absolute

temperature, T, can be represented by

s = B exp (- %%)on (2-25)

32

where R is the gas constant, and B, n and Q are all
constants. He found n equals about 3.2, Q equals 32 Kcal/

24 if the stress ismeasured in

mole, and B equals 7 x 10
bars and the strain rate in years-1. The activation
energy, Q, equal to 32 Kcal/mole found by Glen is higher
than that found by other investigators. Landauer (1957)
pointed out that this value is strongly dependent upon the
use of one creep Specimen at -12.8°C, and a liberal error
could be present in the value. Landauer (1957) did not
disprove Glen's value of Q, as he was attempting to relate
the activation energy of snow to ice. For snow, Landauer
found the activation energy varied from 14 Kcal/mole to
27 Kcal/mole.

Glen (1955) observed that polycrystalline ice
seems to follow an Andrade type of behavior in the
transient creep region if one ignores acceleration due to

recrystallization. From Andrade's law

2 = 2 (1 + Bt1/3)eKt

0 (2-26)

where 2 is current length, £0 is original length, 8 and
K are constants, and t is time.
In terms of the logarithmic strain, this relation-

ship becomes

a = 1n (1 + Btl/3) + Kt (2-27)

33
. 1/3 .
or if 8t << 1, equation (2-27) becomes

a = Btl/3 + Kt (2-28)

Glen assumed that the transient portion followed a tl/3

relation. If it was subtracted, then the straight line

became K = 0.01704"2

which showed a stress exponent of 4.2.
Glen (1955) stated that when polycrystalline ice is sub-
jected to creep, its grains may be either slipping rigidly
past each other, with consequent boundary changes (grain
boundary creep) or else accumulating strain inside the
grains. In the latter case, the probability of recrystal-
lization would rise as the internal strain increased, and
new recrystallized material would be continually appearing.
After a long time a steady-state would be reached in which
the hardening caused by the rise in internal strain is
balanced by the softening caused by recrystallization. A
simple dimensional argument could be used to relate this
strain to the grain size, if the functional relationship of
both nucleation and growth rate were known as a function of
strain. The lack of an accelerated stage in creep curves
at lower stresses is due to the fact that these samples

had not accumulated as much total strain as the others, and
would have developed acceleration later if the duration of
test was long enough, or it might be due to recrystallization
occurring too slowly for the final steady-state to be

reached.

34

Steinemann (1954) wrote the power law equation in
the form § = KTn, where I is shear strain, K and n are
constants, and T is shear stress. Using both compressive
and tensile creep measurements on randomly oriented, small-
grained ice crystals, he found a value of stress exponent
n equal to 4.16 at high stresses. The value of n decreased
with decreasing stress. He pointed out that a simple power
law did not adequately fit all of the data. However, he
observed recrystallization during the measurements,
similar to that postulated by Glen (1955). He stated
that recrystallization takes place when crystals are
deformed inhomogeneously above a certain amount. In poly-
crystalline ice, recrystallization takes place during the
experiment if the applied stress is sufficiently high, or
if the time of testing is long enough for the strain to
attain the necessary value. Thus, polycrystalline ice
remains hard by recrystallization. If this recrystalliza-
tion were absent, a marked transition followed by a linear
part could be expected. Rigsby (1961) also found that
under large shearing stresses ice recrystallizes in such
a manner that the newly formed crystals are favorably
oriented for basal slip.

Nye (1953) found a value of 3.07 for the stress
exponent from tunnel-closing experiments in glaciers.
Butkovich and Landauer (1958), measuring shear creep on

various types of polycrystalline ice, found that within a

35

broad band, all the data fell on a straight line when
plotted as log é versus a with a stress exponent equal to
2.96.

At lower stress levels Jellinek and Brill (1956),
performing compressive creep tests on randomly oriented
polycrystalline ice with noticeable porosity, found a stress
exponent of unity and activation energy equal to 16.1
Kcal/mole. Butkovich and Landauer (1960), also performing
compressive creep tests on randomly oriented ice with
noticeable porosity (average density 0.905 gram/cm3) and
on large grain columnar ice with no porosity, obtained
values between 0.86 and 1.15 for the stress exponent, and
a value of 14.3 Kcal/mole for activation energy. Bromer
and Kingery (1968) performed tensile creep tests with

temperatures from -3° to -13°C at stresses between 2 x 105

dyne/cm2 (2.9 psi) and 2 x 106 2

dyne/cm (29 psi) on
millimeter range grain-size polycrystalline ice. They
analyzed their data by using five theoretical models:

(1) grain-boundary sliding by Kauzmann (1941), (2) grain-
boundary migration by a diffusional process involving
atomic or molecular transfer, (3) microcreep as described
lay Chalmers (1936), Cottrell and Jaswan (1949), and Friedel
(1964), (4) diffusional creep as described by Nabarro
(1948) and Herring (1950), and (5) diffusional creep with

grain-boundary diffusion rate controlling as described by

Coble (1963). They developed the following equation:

36

m
ll

Ac (ss)-m (2-29)
or
—(1> m '
n - 3: (GS) (2'30)

where s is strain rate, 0 is stress, GS is the grain-size,
n is the viscosity, and A and m are constants. They found
that the Nabarro-Herring model provided the best fit for
their data. Their results showed that polycrystalline ice
behaves in a viscous manner, i.e., the strain-rate is
proportional to the grain size squared. The flow process
has an apparent activation energy of 12 Kcal/mole. Mellor
and Testa (1969) following Bromer and Kingery (1968),
performed uniaxial compressive creep tests on fine-grained
polycrystalline ice at a temperature of -2°C and a stress

range between 0.1 and 0.5 Kgf/cm2

(1.45 psi and 7.25 psi).
(Their tests were carried for a total duration of 10.5
months. They showed that the stress exponent of a simple
power relation between strain-rate and stress was 1.8.
Therefore, the earlier results by Bromer and Kingery (1968)
suggesting linear viscous behavior at low stress were
believed to be invalid. Their experiment was simple but
the results were excellent.

The stress exponent and activation energy obtained
at lower stress levels are quite different from those

obtained at higher Stress levels. It appears that there

must be more than a single mechanism reSponsible for

37

plastic flow in polycrystalline ice. Glen (1955) has
‘pointed out that there might be a transition from inter-
granular to intragranular deformation. Butkovich and
Landauer (1960) also indicated that there might be two
mechanisms operating when they proposed a flow law having

a linear and a cubic term in the form
. I 3 _
Y = at + b1 (2-31)

where 4 is shear strain-rate, T is shear stress, a and b
are constants. Dillon and Andersland (1966) in their com-
pressive creep tests on polycrystalline ice concluded that
a transition from one mode of creep behavior to another
appears to separate the high and low stress regions.

As mentioned previously, by theoretical considera-
tions, temperature has a strong influence on the creep
rate of polycrystalline ice. Most theories arerased on
the rate process theory. Jellinek (1960) performed experi-
ments which imposed plastic deformation of thick-walled
snow-ice cylinders under hydrostatic pressure. His idea
basically followed the model presented by Kauzmann (1941).
Dillon and Andersland (1966), applying the rate process
theory, analyzed their results on the creep of cylindrical
polycrystalline ice and proposed an hyperbolic sine stress
and steady-state strain-rate relationship. They proposed

an equation in the form

38

' T
Yoct. = Yo sin h -%SEL (2-32)
0 _

where Ioct. and Toot. are respectively the octahedral

shear strain rate and shear stress, and (o and T0 are
constants. At low stresses, values of (o vary slightly
with temperature. However, the creep curves shown in their
paper had a maximum duration of only 80 minutes and a

total axial strain of 2 per cent. This may be too short

a period or too small a total strain to establish the
minimum creep rate in their range of stresses and tempera-
tures.

Goughnour and Andersland (1968), using experimental
apparatus similar to that used by Dillon and Andersland
(1966), examined the creep oprolycrystalline ice through
the use of compressive tests at high stress. They found
that the creep rate of polycrystalline ice depends on the
stress, temperature, strain and the absorbed strain energy.
Based on their data they proposed creep equation of the

form

to
II

K1 exp (-nls) + K2 exp (n2 f ode) (2-33)

and

(0
ll

K1 exp (-nlt) + K2 exp (nzt) (2-34)

39

where e is the axial plastic strain, 0 is the axial stress
difference, t is the time, and K1, K2, n1 and n2 are
conStants that depend on stress and temperature. In these
equations they assumed that the total effect of all
mechanisms contributing to a decreasing creep-rate could
be represented by one term which they called the hardening
term, the first term in both equations, and that the total
effect of all mechanisms causing an increasing creep-rate
could be represented by the second term which they called
the softening term. These equations appeared to describe
the axial creep behavior of polycrystalline ice throughout
primary, secondary and tertiary creep for strains up to
about 5 per cent and the ranges of stress and temperature
they considered. From the results they indicated that
equation (2-34) fitted their data better than equation
(2-33). Their creep tests were carried to strains of only
5 or 6 per cent. This strain level may again be too small
to adequately demonstrate tertiary creep behavior.

During the course of tests to investigate crack
deformation in polycrystalline, columnar grained ice sub-
jected to a constant compressive loading, Gold (1963)
observed several deformation mechanisms. Some of these
have been described by earlier investigators. The deforma-

tion mechanisms as described by Gold are:

40

1. Slip Bands. Near the melting points single
crystals with hexagonal symmetry usually deform by slip
(on the basal plane. The slip lines define the trace of the
basal plane at the ice surface.

2. Grain-Boundary Migration. Grain-boundary
migration was one of the first signs of change in the
grain-boundary configuration. Every boundary observed
showed evidence of migration. Gold felt that grain-
boundary migration was necessary to relieve the constraints
imposed by the grain geometry.

3. Kink Bands. Kink bands formation is a
mechanism by which a bending moment, transverSe to the slip
direction, can be relieved in crystals with only one or
two possible slip directions.

4. Distortion of Grain Boundaries. As the deforma-
tionincreased, the boundaries of most grains became
severely distorted.

5. Crack Formation. Crack formation is time
dependent as well as stress dependent. The formation of
the crack may be aided by the growth under the applied
Stress of potential crack nucleation sites such as cavities.
Once the crack has formed, the grains associated with it
can more easily conform to the applied deformation. For
creep under severe deformation these can be considered as

"accommodation" cracks.

41

6. Cavities. The cavities were observed in the
region of grain boundaries, grain-boundary triple points,
and the intersection of slip planes and subboundaries.
}Sometimes cavity formation was great enough to form a con-
tinuous column. There was evidence that cavities could
serve as nuclei from.which cracks could develop.

7. Recovery and Recrystallization.

8. Unusual Creep Behavior. Gold found that ice
under compressive creep sometimes exhibited an initial
increasing creep rate. This is unusual creep behavior.

The deformation mechanisms observed by Gold are in
general agreement with the scheme proposed by Shoumsky
(1958). Gold's experimental results suggest that for a
given temperature, creep rate is probably the major
factor in determining what mechanisms are required for a
grain to conform to the applied deformation. At low creep
rates grain-boundary migration, subboundary, Kink band,
and Cavity formation, polygonization, and bending are
likely to be sufficient. For higher creep rates accommoda-
tion cracking during the primary stage and greater dis-
tortion in the grain boundary region occurs. In addition
to creep rate, the amount of deformation and the duration
ofloading are probably important also.

It appears that the deformation of simple crystals
proceeds primarily by means of basal slip; however,

evidence of nonbasal slip has been observed as described

42

earlier. Polycrystalline ice presents a much more compli-
cated picture. At low stresses ice may behave in a quasi-
viscous manner. At high stresses a different mechanism
seems to dominate. Several suggestions and proposals have
been made to account for the flow of polycrystalline ice.
At the present time, none of the flow laws seems entirely
satisfactory throughout the primary, secondary and
tertiary stages of creep. As evidenced by the onset of
tertiary creep, extension of the creep to higher deforma-
tions leads to recrystallization and an apparent softening

or structure change of the ice sample.

CHAPTER III

THEORETICAL ANALYSIS

Development of theory to explain the time dependent
deformational behavior of polycrystalline ice, based on
the observed behavior of ice single crystals will be pre-
sented in Chapter III. The theory will first be developed
for uniaxial deformation and then extended to include
multiaxial creep states.

The experimental portion of this study included
uniaxial creep tests on solid, cylindrical samples, and
multiaxial creep tests on thin-walled, hollow, cylindrical
samples. Equations for computing the plastic strains for

these samples are also derived.

Uniaxial Creep of Polycrystalline Ice
It appears from the previous discussions that there

are two flow regimes for polycrystalline ice. One appears
to dominate at high stresses and it exhibits three stages
of creep. Thus far, it is not clear how the flow rate
depends on the stress under these conditions. The second
mechanism becomes important at lower stresses and will be
discussed with experimental result which may account for

the observed flow behavior in polycrystalline ice.

43

44

In order to develop an equation which will predict
the plastic behavior of ice it is necessary to work with
data that represents plastic deformation only. The
theories presented in this study deal only with plastic
deformation.

Creep such as that illustrated on Figures 6-1 and
6-4, may be characterized as consisting of two parts. The
first part 61, limited to some fixed value, represents
primary creep and may be due to the increasing importance
of grain boundary effects. The second part 62, represents
creep with grain boundary effects fully developed, and the

magnitude of the strain-rate e for a given stress would

2:
depend primarily on the structure of the crystals. A
schematic representation of these mechanisms is shown on
Figure 3-1.

Figure 3-2 is used to explain these mechanisms in a
microscopic sense. The polycrystalline mass shown is
initially relatively stress free. When shear stress is
applied, grain A is oriented for easy glide, while grain
B is not. Consequently as grain A deforms stress concentra-
tions deve10p at the grain boundary in the area indicated,
and a larger portion of the applied shear stress is trans-
ferred to grain B. Since grain B is not oriented for
easy glide the overall creep rate of the polycrystalline

mass decreases. For deformation to continue, mechanisms

such as those described by Gold (1963) are necessary to

45

 

 

 

 

 

 

Figure 3-l.--Schematic Diagram of Creep Curve
Characterized as Two Parts

Basal plane orientation

  
 

Stress concentrations in this area

Grain B

 

\ ‘1'
\ 4——— Applied shear stress

 

Grain A

 

 

 

 

 

 

Figure 3-2.--Stress on Ice Crystals in Polycrystalline
Mass.

46

relieve these stress concentrations. One of these
mechanisms is recrystallization. Recrystallization will
affect softening since the newly recrystallized material
is initially stress free. This is essentially the same
mechanism as described by Gold (1963).

The time law of creep during the course of deforma-

tion as demonstrated above may be expressed as

(3-1)

Recall equation (2-12), the general creep equation

derived by using the rate process theory,

.=fl é—S_ —é—Ii ' TAR «-
Y 2h exp (It) exp ( RT) Sln h —%kT (3 2)

and assume that the mean values of T and § are respectively

proportional to the macroscopic octahedral shear stress

Toct’ and octahedral shear strain rate Yoct.’ For un1ax1al
loading,
_ fz’ . _ .
Toct. — -3'0' and Yoct. - J5 E

where o and é are the applied axial stress and the axial
rate of deformation respectively. Equation (3-2) can be
rewritten in terms of these measurable'quantities, o and E,

as

47

e = /2 %§ exp (%3) exp (- %%) sin h (%;w%%% 0) (3-3)

Dorn and his coworkers (1954) investigated the variation
of the structure parameter, Sp, with strain during creep
at high temperature. They represented the creep rate of

polycrystalline aluminum as

'- -éfl- _
e — Sp exp ( RT) Sln h BO (3 4)

If equation (3-4) is equivalent to equation (3-3), then

S = /2 31 exp (%§)

p h
and
B=Qfl
13 2kT

For constant stress and temperature it seems
reasonable to write either equation (3-3) or equation

(3-4) as

«I
ll

2 SKZ' (3-5)

where K2' is constant for a given stress and temperature,
and S is defined as a structure term. This structure
term may include such factors as softening of individual
grains due to accumulation of strain along their basal

planes (Steinemann, 1954), grain boundary creep, change

48

of grain size during creep, and other factors associated
with ice deformation.
nt

Substituting él’ which is equal to Kl'e- , as

proposed by Goughnour and Andersland (1968), and 62 from
equation (3-5) into the time law equation (3-1), one

obtains

' _ . -nt , _
e — Kl e + K2 S (3 6)

Glen (1955) has discussed a similar argument (see
page 33 ) as described by Figure 3-2. Steinemann (1954)
also stated that if there were no recrystallization, then
even after a slight macroscopic shear, some favorably
oriented grains would be sheared by a sufficient amount
to be soft, so creating a heterogeneous aggregate. The
aggregate itself always constitutes an impediment to
absolutely free glide of the grains and so clusters of
soft grains would be the next stage. An effect of this
sort may account for the observed accelerating stage
(tertiary creep). On introducing recrystallization, an
aggregate can remain hard if the recrystallization takes
place in sufficient time to prevent the large glide,
necessary for softening. If this recrystallization were
absent a marked transition followed by a linear part

would be expected.

49

If one assumes that softening of crystals within
the polycrystalline mass is the dominant factor controlling
the structure term, and S is defined to be the decimal per
cent of softened crystals within the polycrystalline mass,

then
01831.

The rate of softening can be expressed as

' (S - So) (3-7)

—. = ' . - -
K 82(1 8) K4

where K3' and K4' are both constants for any given stress
and temperature. So is constant and equals to S initially
at time zero. The first term of equation (3-7),
K3' 62(1-8), represents the assumed softening behavior,
and the term K4' (S-SO) represents the effects of
recrystallization.

Substituting equation (3-5) into equation (3-7),

one obtains

2

_d_§ _ _ g I l I _ I ' .-
dt _ K2 K3 5 + (K2 K3 K4 )8 + K4 So (3 8)
or rearranging

a? 1 2 3

50

where Cl', C2' and C3' are defined as

Cl — K2 K3 (3 10)
C2 K2 K3 K4 (3 11)
I _ I .—
and C3 — K4 80' (3 12)

K1' and K2' in equation (3-6) are defined as
positive values. By definition K3' and K4' in equation
(3-7) are also positive. Thus Cl' will be negative, C2'
can be positive or negative, and C3' must be positive.

After separating variables in equation (3-9), one obtains

dS

I I
2 S + C3

 

2 = dt. (3-13)‘
c 's + c

l

 

m = /(C2')2 - 4C1'C3' must always be real, therefore,

integration of equation (3-13) gives

2C '8 + C ' + m
i In 1 2 = t + c' (3-14)

I I
2Cl S + C2 + m

where C' is the integration constant. Equation (3-14)

can be written as

I I _ _ I _
2Cl 81+ C2 + m = e mt mc = -ce mt (3_15)

I I _
2C1 S + C2 m

 

51

_ I
where -C = e mC . The choice of signs was made so that

empirical values of C and m will be positive. It is
apparent that C' must be complex.

Solving for S in equation (3-15), leads to

 

I ..
2C ' . -mt
1 2Cl (1 + Ce ).

As time approaches infinity, 8 approaches a steady-state

or equilibrium value of

C2' + m
S = - -———-—-— (3-17)

Now, substituting equation (3-16) into equation (3-6), one

 

 

obtains
I _ '
1 2 2C ' . -mt
1 2C1 (1 + Ce )
or
C ' -mt
° - t 2 n1 1 - Ce
8 = K 'e n - K . - K ' -——T ( _ )
1 2 2Cl 2 2Cl 1 + Ce mt .

Redefining the constants of the above equation,

-nt 1 - ce'mt
( t)

+ K + K _m <3-18)
l + Ce

1e 2 3

52

where

_ I -
Kl _ K1 , , (3 19)
C I
K2 =-K2' 35%7, and - (3-20)
K =-K ' “‘ (3-21)
3 2 EEIT'

By integrating equation (3-18), one obtains

K 2K
1 e nt + K t + K t + -53 In (1 + Ce

2 3

s = -

+ C

7? s

where C5 is the integration constant. And again redefining

constants‘as

- _£ _
C1 — 11 , (3 22)
C2 = K2 + K3, and (3-23)

2K3
C3 = '—m-, (3-24)
one finds

_ _ -nt -mt _

e - cle + Czt + C3In (1 + Ce ) + cs. (3 25)

Equation (3-25) is the final creep equation which is
developed by the concepts described earlier. A series of
tests were performed to test the applicability of equation

(3-25). All parameters appearing in equation (3-25) can

53

be determined for any particular test by the computer
least squares best fit method. Therefore, it is presumed
that all parameters are known values.

If at the beginning of the test, the strain and

time both start from zero, equation (3-25) becomes

-nt mt

e = (—c e -+ci) + c tnt[C3In.(1rtCe- ) - C3In (1 + C)]

1 2

where ' (3-25a)

c - C3In (1 + C) = c

1 5'

If one takes the derivative of equation (3-25),

the creep-rate will be given by

 

é = P e"nt + P + P e-mt (3-26)
1 2 3 l + Ce—mt
where
P1 = Cln,
P2 = C2, and
P3 = -C3Cm.

Equation (3-26) has the same meaning as equation (3-18)
except all of the parameters appearing in equation (3-26)
are known. Therefore, one can calculate the creep-rate

at any time during creep.

54

From equations (3-24), (3-21) and (3-10), one finds

K I

Since C3 is known, K ' is known also. From equations

3
(3-23), (3-20), (3-24), (3-10) and (3-27) one obtains

 

I I
C=K+K=-K'C2+..—.—C3m.-=CzC3+——C3}m
2 2 3 2 2Cl' 2 2 2 '
This leads to
.2C2
C2' = IT— - m. (3-28)
3

The value of C ' can be obtained from known values of C

2 2.

C3 and m.

In order to evaluate the constants, K2' and K4',
the experimental procedure illustrated in Figure 3-3 is
necessary. The sample is loaded to some stress which is
maintained until time t1. The sample is then unloaded
and allowed to "rest" until time t2, at which time the
original stress is reapplied and creep continues. The
first loading will be referred to as Phase I and the
second loading as Phase II. Phase I is equivalent to a
conventional creep test, and the parameters of equation

(3-25) can be fit to Phase I test data, in the usual

manner. Hypothetically equation (3-25) is valid for both

55

Unload Reload
I I

   
    
 

Phase I Phase II

Strain ———"

.‘_J&EEL2£M;_;J
I

 

 

 

 

Figure 3-3.--Schematic Diagram of Two Phase Loading.

Phase I and Phase II. Consequently the parameters of
equation (3-25) can also be fit to Phase II test data but
these parameters will represent different initial condi-
tions (80 value).

At the beginning of loading in the first phase

(t = 0), equation (3-16) can be written

 

_ _ _ 2 _ m (l - C) _
S “ So ‘ 25:7 2c1' (I*+ C)’ (3 29)

Multiplying equation (3-29) by KZ', and from equation
(3'27) I

56

. __2_ m(l-C)___g_ _
K2 So “ 2 C3 + C3 E‘TI‘I'ET" 2 C3 + B (3 30)
where
B = C m'(1 - C)

 

3 2 (l + C)’

Substituting equation (3-28) into equation (3-30), one

obtains

 

(3-31)

where KZ'So can be computed from known Phase I values of

C C3, C and m.

2'
During the rest period, while recrystallization is

occurring, equation (3-7) should be written as

= - K

010--
('1’ U)

4' (S - So). (3-32)

Integration of equation (3-32) gives

In (so - S) = -K4't - c"
or
so _ s = e-K4't-C" = _ C'"e-K4't
where C"' = -e-C" is constant. It follows that

57

-K4't
S = C"'e + So. (3-33)

Referring to Figure 3-3, when t equals t1, S equals S

Then

- K 't
III = _ 4 1

Substituting back into equation (3-33), one obtains

-K4'(t-t1)
S = (Sl - So) e + 50' (3-34)

When the rest time reaches t2, and S = $2, equation (3-34),

multiplied by K2', becomes

. , - 4'(:2-:1)I ,
— ' - -

where t2 - tl is the rest time. K2' was originally

assumed to be stress dependent, but since the stress is

the same during the first and second phases, K2' is the

same during the first and second loading phases.

Now, from equation (3-16), one finds the expression

of S in Phase I at t = t

1 1

58

 

 

. K '
Since from equation (3-27), C3 = - CETV multiplying by
' 1
K2' yields
-mt
C ' I , .V ..... 1
K2'51 = '2' C3 + C3 m (l ce-mE)
2 (1 + Ce 1)
or it reduces to
C2'
I - _
K2 Sl — -§—'C3 + D (3 36)
where
-mtl
_ m (l - Ce )
2 (1 + Ce 1)

If one substitutes equation (3-30) and (3-36) into

equation (3-35), the result is

-K '(t -t )
Kz's2 = (D-B)e 4 2 1 + -§ c ' + B. (3-37)

Equation (3-37) expresses a relationship between Phase I
and Phase II.

If one considers time starting from zero in the
second phase, it is possible to obtain best fit parameters
of equation (3-25) for this data. Then for time zero for

the second phase, equation (3-16) yields

59

K2 S2 2 C3 + C3 2 1+C) 2 C3 + B2 ‘3 38)
_ In (l-C)

Note that all parameters in equation (3-38) are result of
Phase II test data. Equation (3-38) is equivalent to

equation (3-37), thus

 

C' -K'(t-t) C'
2 _ _ 4 2 1 2
—-2 C3|II+B2—(DB)e +——2 C3II+B
(3-29)
where the subscripts I and II indicate the Phase I and
Phase II parameters. B2, D and B are defined as pre-
viously. Since all parameters except K4' are known in
equation (3-29), one can solve for K4', which gives
C ' C '
2 2
—-—cl --—c|+B-B
K4' = t it In 2 3 II 2 3 I 2 (3_40)
1 2 D - B
As long as K4' is determined, Kz' can be obtained from
equations (3-11), (3-27) and (3-28), which gives
I_. - .—
K2 — K4' C3 + 2C2 C3m. (3 41)

60

K2' expresses the stress dependence of equation (3-5).

More details of K "will be discussed with the experimental

2
results.

Once K2' is obtained, Cl' can be found from
equations (3-10) and (3-27). Thereafter the value of the
structure term S can be found from equation (3-16) for any
desired time. One can also obtain the constant So from

equation (3-12) in which C ' can be found from

3

 

— 2-
m - /(c2') 4 cl'c3'

It is realized that So will be very small compared to S.

As noted previously, when time approaches infinity,
S approaches some equilibrium value. Equation (3-5) is
then at the final creep rate stage, and equivalent to C2

in the equation of ‘
‘ /

This should be equivalent to equation (3-17).

Extension to Multiaxial Creep

 

In most analyses in Continuum Mechanics, two
assumptions about the nature of the material are made,
namely, that it is homogeneous, and isotropic. It should
be clearly understood that there are actually three com-

pletely independent assumptions. A material is continuous

61

if it completely fills the spaces that it occupies,
leaving no pores or empty spaces. A homogeneous material
has identical properties at all points, and a material is
called isotropic with respect to certain properties if
these properties are the same in all directions.

The determination of the continuum concept of
matter for ice has to be investigated. A thin-walled,
hollow, cylindrical ice sample was considered for this
study. This model can be applied to three dimensional
stress states.

At any point in the material, assumed to be homo-
geneous and weightless, let Oij be the components of
stress, and Ui be the components of flow. The equations

of slow motion of an element are then

80.

fi=O(i.j=l,2.3)

where the summation convention is used for repeated
suffixes (Nye, 1953). The rate of strain tensor at any

point has components

, 1 3U]: 3U.
Eij = '2' (3;; + ‘43in- (3'43)

For incompressible material, i.e., 81 + 62 + £3
= 0, and isotrOpic throughout the flow, the principal

axes of éij must be parallel to those of Oij and the

62

components of the strain rate proportional to the

components of the stress deviator, thus

8.. = 10!. (3-44)
13 13
where Gij' the stress deviator, is given by
0 =0 --1-3..c (3-45)
ij 13 3 13 kk

and A is a positive scalar factor of proportionality that
depends on position and time. The term %.aij Okk is

called the spherical stress tensor, and can be considered
to act as a hydrostatic stress. In plastic flow considera-
tions one can consider the stress system obtained by sub-
tracting the spherical state of stress from the actual
state, rather than working with the actual state of stress.
There is furthermore evidence that the hydrostatic pressure
does not affect the flow rate of ice (Rigsby, 1958).

In three-dimensional stress states, the shear
stresses and shear strains on the octahedral plane give a
good representation of the shear distortion in the
specimen. If experimental stresses and strains are taken
under the principal directions, they can be converted to
octahedral stresses and strains. The theoretical creep
equation as derived in equation (3-25) must be modified
then to be applied to the thin-walled, hollow, cylindrical

samples.

63

The octahedral shear stress is given by

 

 

_ 1 _ 2 _ 2 _ 2 _
Toct. - 3 H01 02) + (02 03) + (a3 cl) (3 46)
where 01, 02 and 03 are three principal stresses, and
octahedral shear strain is given by
_ 2 _ 2 _ 2 _ 2 _
Yoct. — 3 /(e1 :2) + (62 63) + (e3 61) (3 47)
where 81, £2 and 83 are three principal strains. In
uniaxial loading, equation (3-46) becomes
_ J?
Toct. - IT 0' (3 48)
and equation (3-47) becomes
7 = /2 e I (3-49)

oct.

where c and 5 denote the applied axial stress and the
axial strain respectively.

In order to convert equation (3-25) from the
principal sense into the octahedral sense one must multiply
by a factor of /2 and convert the stress dependent param-

. 3 .
eters from 0 into /E T . The result 13

oct.

64

-noct t
eoct. = -Cl oct. e + C2 oct.

-moct.t

(1 + Ce ) + c5 oct ' (3-50)

One can determine the best fit parameters of equation

(3-50) for experimental data by using the same technique

as described previously. Theoretically, these parameters
determined from the results of tests on thin-walled, hollow,
cylindrical samples should be identical to the results of
solid, cylindrical samples. The comparison between these

two results will be given later with experimental results.

Plastic Strain Computation

 

Solid, Cylindrical Sample

When a solid is subjected to a static, axial force,
the atomic lattice will adjust itself to oppose the
applied force and maintain equilibrium. On a macroscopic
scale the atomic adjustment is observed as deformation
when the lattice remains continuous and as fracture when
the atoms are pulled apart. Deformation involves displace-
ments which when sufficiently large can be measured
macroscopically. The magnitude of these displacements
depends on the dimensions of the element deformed. Dis-
placements cannot be compared directly unless referred to
unit elements and thus converted into a dimensionless

quantity called "strain." Under a simple tensile or

65

compressive force the strain can be directly computed

from displacement, or change in length, in the direction

of applied force. Often the strain is computed as a normal
strain E, which refers the change in length, dk, to an

original length, 20, and is given by

_ 2
. =I Tet—l =37 (3-51)

The displacement also can be converted into a true strain,
a, which refers to the change in length to the instantaneous

length and is given by
z _
From equations (3-51) and (3-52), one obtains

e = III (1 + E). (3-53)

For compression, equation (3-53) becomes

m
ll

- In (1 - E)

or

- In (1 - A31). (3-54)

00
ll

66

If one only measures the plastic deformation of the speci-
men, subtracting elastic deformation and system error
during the course of the experiment, equation (3-54) can

be written as

. A9; ._. C

ILo

e = -In [1 - ] - G (3-55)
where C is elastic deformation, and G is the system error
which is calibrated from the test equipment. Equation
(3-55) was used to compute the plastic strain for all
uniaxial compressive creep tests in this study.

Thin-Walled,iHollow,
Cylindrical Samples

 

 

When pressure is applied internally to a thin-
walled, hollow, cylindrical sample, the experiment is con-
sidered to be a tensile test since the specimen is
radially expanded. The conventional strain can be

written as

r.- = —fi—2 = g2 (3.56)
o o
where D0 = initial inner diameter of the hollow
cylinder, and
D = instantaneous inner diameter of the hollow

cylinder.

67

The initial volume and current volume can be expressed

 

as
nDoz
v0 = 4 hO ‘ (3-57)
2
_ flD _
V _.Tho (3 58)

Where ho is the height of specimen. Substituting equations

(3-57) and (3-58) into equation (3-56) yields

_ = (_y_)1/2

5 v
0

_ 1 (3-59)
Applying equation (3-53), the true strain becomes

a = %- In (3-60)

o<I3

For compressive creep tests, the pressure is
applied to the specimen as a confining pressure. In a

similar manner, the true strain becomes
V
_ l o _

Equations (3-60) and (3-61) were used for computing
the plastic strain for appropriate tests. Note that the
initial volume of the inner hollow cylinder was taken

after elastic displacement had occurred.

68

The tensile or compressive hoop stress in the thin-

walled, hollow, cylindrical ice sample is obtained from

the equation .

_.E£ _
St - 't (3 62)

where P is the applied hydrostatic pressure, r is the
inner radius in the case of internal pressure and the

outer radius in the case of external pressure, and t is

the wall thickness.

CHAPTER IV

SAMPLE PREPARATION

Methods of preparing ice samples are described in
Chapter IV. General information pertaining to both solid,
cylindrical samples and thin-walled, hollow, cylindrical
samples includes use of only distilled, deaerated and
deionizEd water; freezing temperatures; and aging history.

Differences in samples due to aging were minimized
by using the same aging history for all samples. The
samples were left in the mold in a cold box, maintained at
a temperature of -18 i 2°C for about 24 hours. They were
then encased in rubber membranes and mounted in the test
cell where they remained submerged in cooled ethylene

glycol-water mixture for another 18 hours before testing.

Solid, Cylindrical Ice Samples
Cylindrical samples 1.13 inches in diameter by
2.26 inches high were prepared in a four compartment
aluminum mold. The diameter of 1.13 inches was selected to
give an initial cross-sectional area of one square inch.
After each use the mold was taken apart, cleaned,

reassembled, and coated inside with Dow-Corning, high

69

70

vacuum silicone grease which prevented leakage and pre-
vented adhesion between the mold and the ice.

The cylindrical polycrystalline ice samples were
obtained by filling the chilled mold with loose, dry,
clean natural snow, adding precooled high-purity water
into the snow, and allowing the mass to freeze. Virtually
none of the snow melted when the precooled water was
added. Freezing was accomplished in the cold box main-
tained at a temperature of -l8° i 2°C. Nucleation was
observed to proceed through the sample at a moderate rate
and during the following growth the samples were not
disturbed. The frozen samples were left in the refrigerator
box for approximately 24 hours before sample tops were
trimmed and the samples were removed from the mold. Before
mounting samples in the test cell, care was taken to
discard any samples which contained excessive bubbles or
cracks, or which appeared to have a large variation in
density and strength.

The resulting polycrystalline ice samples were
cloudy in appearance, and measured densities ranged
between 0.900 gm/cm3 and 0.905 gm/cm3. There was a slight
radial pattern visible in some samples when they were
broken apart. In general most samples were formed quite
homogeneously. Examination of thinly cut sections under

a microscope revealed an irregular fine-grained aggregate.

71

Mean grain size was about 1.0-1.5 millimeters in diameter.
The overall distribution of the orientation of the indi-

vidual crystals is believed to have been random.

Thin-Walled, Hollow, Cylindrical
Ice Samples

The mold used to form the thin-walled, hollow,

 

cylindrical samples consisted of one piece of plastic pipe,
with inner diameter to match the outer sample diameter,
seated and mounted on an aluminum base plate by four
screws. A four piece collapsible aluminum core was secured
at the center of the plastic pipe by single steel screw.

The disassembled mold is shown in Figure 4-1.

 

Figure 4-1.--Disassembled Mold for Forming Thin-
Walled, Hollow, Cylindrical Ice Samples.

72

Before assembling the mold, a thin coat of high-
vacuum silicone grease was spread by hand as thin as
possible on the plastic pipe inner face and on the outer
face of the collapsible aluminum core. A thin polyethylene
film was then placed over these surfaces. The silicone
grease served as an adhesive to hold the polyethylene
film in place. This technique was found to facilitate the
removal of the collapsible aluminum core from the inside
of the sample and also to facilitate removal of the sample
from inside the plastic pipe. The assembled mold was
then placed in the refrigerator box for two hours. It
was found that two hours was the best time to cool the mold
for making ice samples. If the time was too short, the
mold heat would melt the filling snow, while if the time
was too long, the mold would become too cold and freeze
.the precooled water immediately during pouring without
saturating the filled snow.

Since the sample thickness was so small, it was
difficult to fill fresh snow into the small gap in the
mold. To prevent this difficulty, the snow was stored
in the refrigerator for several days until it condensed
to small fine particles. The fine-particle snow behaved
similar to fine grained sand and filled the mold adequately.
A precooled high-purity water was carefully poured to
infiltrate the snow. The mixed slush was then frozen to

form polycrystalline ice.

73

In order to reduce the number of air bubbles in
the sample, the mold was slightly vibrated while the water
was poured into the snow. A completely air-free sample
was very difficult to obtain, and many samples had to be
discarded before an acceptable one was obtained. Measured
densities were about 0.910 gm/cm3, and the grain size was
approximately l.5-2.0 millimeters in diameter. A photo-
graph of a thin-walled, hollow, cylindrical sample is
shown on Figure 5-7.

Care was necessary in ejecting the ice samples
from the plaStic pipe. If the sample was ejected too
rapidly cracked layers appeared in the ice. Moderate
pushing force proved best to avoid such damage.

The measured size of the thin-walled, hollow,
cylindrical ice sample was typically 1.996 inches in
height, 2.089 inches outer diameter and 1.801 inches
inner diameter. The wall thickness was 0.144 inches.

For this size ratio, the samples were considered to be

thin-walled.

CHAPTER V

APPARATUS AND EXPERIMENTAL PROCEDURES

Experimental procedures and apparatus used for
tests on solid, cylindrical samples were similar to those
used by Goughnour (1967). A modified triaxial cell was
used for tests on hollow, cylindrical ice samples. The
axial loading system was similar to that used for the
solid, cylindrical sample tests.

All samples were kept in their molds in the
refrigerator box for approximately 24 hours before the
sample tops were trimmed flush to the top of the molds.

The samples were then removed from each mold and weighed.

For all samples, friction reducers were used at
each end between the sample and the bearing plugs. The
friction reducers were made by coating one side of aluminum
foil with silicone grease, spread as thinly as possible by
hand. A thin film of polyethylene was then applied to the
greased side of the aluminum foil and air bubbles were
worked out by hand. These sheets were then cut into
disks of the prOper size to match the ice samples. A
previous investigation (Goughnour, 1967) found such friction

reducers to be very efficient.

74

75

Rubber membranes were then secured over the
samples by rubber bands. Detailed information for further
preparation is given under headings for solid, cylindrical
ice samples and hollow, cylindrical samples. Information

on testing procedures is also included in this chapter.

Solid, Cylindrical Ice Samples
The triaxial test cell, lucite disks and the test
sample are shown on Figure 5-1, and the loading setup is

shown schematically on Figure 5-2. Friction reducers and

 

Figure 5-l.--Triaxial Cell, Test Sample, and Mold
for Solid, Cylindrical Ice Samples.

76

Connected to Electrically
Powered Jack

 

I

 

49.4.1

If

Loading Frame

LI”

 

Thermometer

 

Coolant out

 

IT

 

 

 

 

 

 

 

 

 

Membranes

 

II 1111']

 

Coolant in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

System.

  

 

 

 

 

 

 

~““~.I 1 Strain Gage E
= a Recorder \. . 5 Coolant
E E gL 2 ;,’-(Ethy1ene
= _lI__ —-=-- _‘-’-' d, /: Glycol and
Lucite /'Rubber 3 : Water)
{ disk - bands 9 : Triaxial Cell
8‘ I u
: ~§ _ I' =.
’ Ice I: F
To \ Sample v,,¢~1"S Tank
Recor er .1 =
3: ri— Insulation
{ {a E (styrofoam)
1". :‘ :
Force TransducerE
5 Loading
Frame

 

4——-—-Dead Load (Lead Bricks)

Figure 5-2.--Schematic Diagram of Axial Loading

77

lucite disks were placed at each end of the trimmed ice
samples, and two rubber membranes were placed over the
sample and disks. The rubber membranes were secured to
the lucite disks by rubber bands. The prepared sample

was then secured in place on the pedestal of the triaxial
cell as shown on Figure 5-2. An all-steel triaxial cell
was used in order to minimize friction between the loading
ram and the bushing due to the cold temperature. This
stainless steel ram was kept lubricated with light oil to
eliminate any possible friction in the ram. Normally, the
ram falls slowly under its own weight prior to the appli-
cation of the load.

The entire triaxial cell was placed in a galvanized'
tank and immersed in a coolant. The galvanized, open t0p
tank was encased, sides and bottom, by heat insulating
styrofoam. The coolant was a mixture of ethylene glycol
and water, and was maintained at a constant temperature by
circulating through a low temperature bath equipped with
a thermoregulator.

Temperatures adjacent to the test cell were measured
by a laboratory type thermometer. Temperature control was
maintained to within 0.1°C. Results of a previous study
by Goughnour (1967) using a thermistor inside the test cell
immediately adjacent to the sample, indicated a temperature
about 0.05°C warmer than that measured outside the test

cell.

78

Axial loads and axial deformations were recorded
by a two-channel Sanborn recorder which provides a con-
tinuous output record. Maximum sensitivity produced one
centimeter needle deflection for 5 pounds of load and
needle deflection of one centimeter per 0.002 inches sample
(displacement. The base line on the chart record made by
the recorder could be shifted as a test progressed making
it possible to use maximum sensitivity throughout the test
if desired.

Axial loads were measured by a force transducer
mounted in the base of the test cell. Ram friction was
thus avoided, and loads were measured with an accuracy of
about 0.25 pounds. Axial deformations, determined by means
of a linear differential transformer, were recorded with
an accuracy of about i0.0025 mm. Measured deflections
included an elastic component contributed by the test
equipment, and a nonrecoverable portion, labeled seating
error, contributed by further compression of friction
reducers and adjustment of the contact between the sample
and lucite disks.

Although special care was taken to trim the sample
ends to a flat surface, a perfect fit against the lucite
disks was not possible. This difficulty was reflected in
the test results and was corrected by a method of pre-

loading. After the sample was mounted in the test cell,

79

a load of 2.1 mm"2

(250 psi) was applied to the sample for
10 minutes and removed. The sample was then allowed to
"rest" overnight before testing.

The elastic deflection of the loading system was
evaluated during a previous study (Goughnour, 1967).
Elastic deflections were found to be 0.0008 inches per 100
pounds of load. This correction was considered here in

the data presented since the test cell was the same one as

used in the previous study. Figure 5-3 is a photograph of

 

Figure 5-3.--Triaxial Cell in Coolant, Test Machine,
Cooling System and Sanborn Recorder.

80

the test machine, coolant tank containing the triaxial
cell, cooling System, and Sanborn recorder.

For the constant axial stress creep tests of solid,
cylindrical ice samples, a load was applied to the specimen
by means of a load frame, which supported a dead load of
lead bricks and was lowered onto the loading ram by a
mechanical loading system (see Figure 5-2). The load was
lowered at a fairly fast rate and it took about 4 seconds
for the entire load to transfer to the sample. Dynamic
effects were assumed to be small, and were ignored. As
the sample deformed, a constant unit stress was maintained
by adding lead shot to the dead weights of lead bricks to’
compensate for the small increase in sample cross-sectional'
area. Constant sample volume was assumed for computing
the required weight of lead shot. No confining pressure
was applied to the sample except for about 6 inches head
of the cooling liquid on all samples.

Small variations in ice densities were assumed to
be due to very small air bubbles in the ice, since no
large voids were visible. It was assumed that no internal
weakening of the ice resulted from these air bubbles and
that the effective area of the samples was reduced by a
.factor proportional to the volume of air contained. Test
results were improved by correcting the computed stress
on the samples by a factor proportional to the variation

of the sample density from some standard density. Data

81

reported have been corrected to a sample density of 0.903

3

Mg m‘ (0.903 gm/cm3).

Young‘s modulus for polycrystalline ice was taken

as 6000 MNm‘2 5

(8.7 x 10 psi), and elastic sample deflec-
tions as well as system deflections have been subtracted
from all data presented. Thus the strains reported here
represent only the plastic natural strain. All tests were
conducted at a temperature of -4.45°C.

Some samples were deformed to strains in excess of
25 per cent. Sample geometry changes, inherent in uniaxial
creep tests, were of course large; this is recognized as
a limitation on the resulting data. The friction reducers,
however, appeared to be very efficient, and samples
remained uniform without barrelling even though strains
were large, as shown by Figure 5-4.

Thin-Walled, Hollow, Cylindrical
Ice Samples

 

 

The test machine, loading frame, coolant tank,
cooling system, and Sanborn recorder used in the solid,
cylindrical ice sample tests (see Figure 5-3), were used
also for the thin-walled, hollow, cylindrical ice sample
tests. A triaxial test cell of the type used in tests on
solid, cylindrical Samples was modified appropriately for ~
tests on hollow, cylindrical samples, and is shown

schematically on Figure 5-5.

82

 

Figure 5-4.--Deformed Solid, Cylindrical Ice
Sample Compared with an Undeformed Sample.

After the samples were prepared in the mold they
were trimmed flush with the top of the mold and then
removed from the mold. Aluminum caps and friction
reducers were then placed at the top and bottom of the
sample. A rubber membrane was placed inside the hollow
cylinder, stretched tightly, and secured by two steel
rings fitted with rubber o-rings (see Figure 5-5). A
rubber membrane was then placed over the outside of the
sample and aluminum caps, and secured to the aluminum

caps by tight fitting rubber bands. This prepared specimen

83

Axial Loading Ram

  
   
      
 
 
 
 
 
  
 
 
  

Valve 4

Valve 3 Top Aluminum Cap

Rubber Bands

Friction Reducers Ice Sample

(Tpr

Rubber
Remembrane

Coolant

Steel "0" Ring .
Bottom Aluminum

Valve 2

Force Transducer
Valve 1

Figure 5-5.--Schematic Diagram of Triaxial Cell
Modified for Testing of Thin-Walled, Hollow, Cylindrical
Ice Samples.

84

was then placed in the test cell with the bottom aluminum
cap screwed in the pedestal of test cell which connected
to the force transducer. The steel ram in the test cell
was then screwed into the top aluminum cap. The entire
triaxial cell was immersed in the coolant tank. The top
cell valve (valve 4) was left open. After the cell
filled with coolant liquid the valve was closed. The
space inside the hollow ice specimen was filled with
coolant liquid by means of valve 1.

The source of the hydrostatic pressure applied to
the sample was a cylinder of compressed nitrogen. The
inner hollow space of the specimen communicated with a
small hole at the center of the bottom aluminum cap which
in turn was connected to the nitrogen cylinder or to a
burette, depending upon the objective of the tests. This
Operation was controlled by valves 1 and 2. If the nitrogen
cylinder was connected to valve 1, the valve, 2, was con-
nected to a burette, or exchanged vice versa.

The hydrostatic pressure from the nitrogen cylinder
was controlled by a pressure regulator placed at the
cylinder outlet, and a gage connected to the pressure
regulator was used to determine the pressure applied to
the test cell. The pressure could be measured with an
accuracy of about i0.1 psi. A pressure hose transmitted
the pressure directly from the nitrogen cylinder to the

teSt cell. The burette was used to measure the contraction

85

or expansion of the inner hollow volume of the Specimen.
The burette and the connecting flexible tube were kept
submerged in the coolant in order to minimize expansion

of the fluid due to any temperature difference. Prior

to testing, the coolant liquid was filled in the connecting
flexible tube up to the zero level of the burette. The
burette was always read with the meniscus level with the
surface of the coolant surrounding the triaxial cell,

thus eliminating corrections needed for any change in

head. The burette was calibrated directly to 0.05 cm3.
A photograph of the nitrogen cylinder with test apparatus

is shown on Figure 5-6.

Elastic system deflections were evaluated by
substituting a steel plug for the ice sample, applying a
load, and measuring the resulting deflections. Elastic
system deflection was found to be 0.0003 inches per 100
pounds. Seating errors were minimized by a preloading
technique as described previously.

For constant axial compression stress creep tests,
hydrostatic pressure from the nitrogen cylinder was not
applied to specimen. Therefore, valves 1 and 2 were closed,
and valves 3 and 4 were open. Axial loading was the only
pressure applied to the Specimen, and the deformation of
the sample was measured in a longitudinal direction.

Constant tensile stress creep tests were performed

by applying pressure from the nitrogen cylinder internally

86

 

Figure 5-6.--Nitrogen Cylinder and Testing
Apparatus.

to the specimen through valve 1. Valve 2 was connected

to a flexible tube and a burette. Valves 3 and 4 were
closed. Because internal sample pressure will uplift the
top aluminum cap, a balancing axial pressure was applied
to the loading frame immediately after the internal hydro-
static pressure was applied. The sample displacement was
computed in terms of the volume of coolant liquid expelled
because of expansion of the ice sample. This volume of

coolant liquid was measured by means of the burette.

87

An elastic displacement of the sample was observed
in the first few seconds after internal hydrostatic
preSsure was applied. Data reading started immediately
after this elastic response.

Since the wall thickness of sample was considered
thin, the stress could be taken as constant across the
wall. This method has been used extensively in the study
of the plasticity of metals. The tangential stress within
the sample wall is computed from the applied pressure and
the section of the sample.

Constant compressive-compressive biaxial stress
creep tests were performed by applying axial load to the
loading frame, while hydrostatic pressure, supplied from
the nitrogen cylinder, was applied through valve 2 onto
the exterior of the specimen. The burette was connected
to valve 1, in this case, in order to measure the contrac-
tion of the sample. Valves 3 and 4 were closed. Axial
load indicated by the load transducer was a result of the
dead load weight plus that caused by hydrostatic pressure
within the triaxial cell.

During the test the axial deformation was measured
by a linear differential transformer and recorded on the
chart recorder, while the lateral deformation was measured
by the burette as the inner volume of hollow cylinder

changed. Corrections for system errors and elastic

88

response were the same as before. A sample deformed under
compressive-compressive biaxial stress, compared with an

undeformed sample is shown photographically on Figure 5-7.

 

Figure 5-7.--Deformed Thin-Walled, Hollow,
Cylindrical Sample Compared with an Undeformed Sample.

CHAPTER VI

RESULTS AND DISCUSSION

Experimental results and discussion are presented
in this chapter. All experimental data presented were
obtained at a sample temperature of -4.45°C. Parameters
in the creep equation were empirically evaluated by best
fitting, in a least squares sense, to experimental data.

The CDC 3600 computer facility was used for this purpose.

Solid, Cylindrical Ice Samples

 

High Stress Level
The definition of high and low stress is still
uncertain. Bromer and Kingery (1968) called stresses below

6 dyne/cm2 (29 psi) low. Dillon and Andersland

2 x 10
(1967) defined the high and low stress regions by axial
stress greater or less than about 85 psi. For this study

2

stresses of 0.686 MNm- (100 psi) or greater will be con-

sidered high, and stresses below 0.686 MNm-2

(100 psi)
will be considered low. Low stresses are those which do
not produce a tertiary creep stage. The magnitude of
this threshold stress is of course temperature dependent.
Also stresses near this threshold value may produce

uncertain results.

89

90

Figure 6-1 shows creep curves for five test samples
(1A to 5A). These data represent results obtained under
a relatively high compressive stress of 2.38 MNm-2
(350 psi). Attention is called to the consistency of the
creep rate in the tertiary portion of creep of each curve.
All five curves display a relatively constant creep rate
of 7.40 x 10-5 sec.1 after having total strain of about 12
per cent. The slight deviation among the curves may be
attributed to small variations in density, impurity of
material, grain size and random orientation of individual
crystals or experimental errors.

Figure 6-2 shows a plot of strain-rate, é, versus
strain, 6, for test sample 43. Notice the final constant
creep rate. This is typical for all series B tests. The
data presented in Figures 6-1 and 6-2 appears to be con-
sistent with the theory presented earlier.

Figure 6-3 displays the results of test sample 6A,
where the sample was subjected to three separate loadings
with intermediate overnight rests. The first rest time
was 7.764 x 104 seconds, and the second rest time was
7.344 x 104 seconds. Each loading was at a stress of

1.715 MNm—z

(249 psi). It appears that recrystallization
during the rest times restored the sample to virtually its
original condition, although gradual granular reorientation
may account for the slightly greater creep-rate attained

during the tertiary portion of each successive creep

91

curve. If recrystallization occurs while the load is
removed, it seems reasonable that it also occurs during
creep. Glen (1955) has postulated this result. Steinemann
(1954) has observed recrystallization in inhomogeneously
strained single crystals and pointed out that recrystalli-
zation took place sometimes during the experiment, some-
times after unloading. Gold (1963) also has observed
evidence of recrystallization at the surface and internally
after the load had been removed. However, no evidence was
obtained from him to show that recrystallization occurred
during deformation.

The apparent difference in the secondary portions
of the curves for the first loading and second loading
shown in Figure 6-3b is believed to be the result of the
seating errors. These seating errors were reduced by a
preloading technique used in all series B tests.)

Figure 6-4 shows the creep curves at several
stresses for series B tests (13 to 9B). The strains
represented in the figure have been adjusted for sample
elastic strain and system deflection according to equation
(3-55). It can be seen that the tertiary portions of

2

creep for the stress range between 0.686 MNm- (100 psi)

2

and 1.365 MNm- (200 psi) reached a constant strain-rate

after about 10 per cent of total strain, while for stresses

2 2

in the range between 1.710 MNm- (250 psi) and 2.740 MNm-

(400 psi), the constant strain-rate was not reached until

92

nearly 12 per cent of total strain. It is believed that
the relatively larger equilibrium value of the structural
term, 8, is responsible for this behavior. This behavior
will be discussed more completely later in this chapter.

Figure 6-5 illustrates the results of fitting the
data for sample 8B to equation (3-25). It can be seen
from the figure that when data up to 25 per cent strain are
included, the residuals are within 0.1 per cent. The
residual at any time is computed as the actual measured
strain minus that computed by the equation. This result
is typical for all tests.

Table 1 lists the parameter values of equation
(3-25) obtained for series B tests, covering the range of
stresses indicated. The preloading technique to reduce
seating errors was not used on series A tests. Consequently
thesetest results were erratic. Series B tests only were
used in the fitting process. The value of C5, the inte-
gration constant, has been omitted from Table 1. Attention
is called to the uniformity of parameter C1. Figure 6-6
shows a plot of C1 versus stress and indicates that Cl is
independent of stress. Experimental error is believed to
be responsible for the random variation of C1. If the
first term of equation (3-25) represents primary creep
behavior, and primary creep behavior is indeed due to the
increasing importance of grain boundary mechanisms, then

Cl represents the total axial strain necessary for these

93

mechanisms to develop fully. Since these grain boundary
mechanisms must be developed fully for all stresses, it

seems reasonable that the magnitude of C should be

1
independent of stress. The average value of C1 was taken
as 0.581 per cent (0.00581).

Figure 6-7, a plot of C versus stress for series
B tests, indicates that C is also independent of stress.
Since the samples for all series B tests were allowed to
come to equilibrium overnight, it was assumed that the
initial value of 8, So' was constant for all tests prior
to loading. The variation of C values may be attributed
to small variations of density, impurity of material, grain
size and random orientation of crystals from sample to
sample, or ascribed to slight experimental errors during
testing. In identical samples, the C values should be the
same for any stress. The average C value of 62.5 was
assumed to represent initial conditions for these experi-
mental conditions.

Because of the large number of parameters in
equation (3-25) it is possible for these parameters to
undergo quite large variations relative to one another and
still produce a good fit to the data. In order to fix the
values of parameters for individual tests more definitely,
the fitting process was repeated with the value of C1
fixed at 0.581 per cent and the value of C fixed at 62.5.

The results are given in Table 2. It appears that C and

94

C1 are material dependent and n, C2, m, and C3 may be
stress and temperature dependent.

The parameters of equation (3-25) were assumed to
be constants for any given stress and temperature. Param-
eters Cl and C appear to be independent of stress for the
range of stresses studied. Plots of the remaining
parameters versus stress for constant temperature indicate
that these remaining parameters are stress dependent.
Figures 6-8, 6-9 and 6-10 show plots of parameters n, m
and C2, respectively, versus stress on logarithmic scales.
Figure 6-11 shows parameter C3 as a function of stress.
Lines of best fit were drawn visually through the data
points on these plots, and the equations of these visual
lines of best fit are given in the figures. Note that

the stress is expressed in .Mega Newtons per square meter

(MNm-2

). One MNm-2 is equal to 145 pounds per square
inch. The relation between parameters n, m and C2, and

s tress are respectively

n = 8.28 x 10'504'85 5.1 (6-1)

5.83 x 10'50 4'05 s"l (6-2)

5
ll

2 1.89 x 10'40 4'22 8s'1. (6-3)

0
ll

. . -2
where o is in MNm .

95

The term, Czt, in equation (3-25) represents the
flow of polycrystalline ice when the structure term, S,
has reached its equilibrium value and grain boundary
effects are fully develOped. Attention is called to the
excellent consistency of these data. The stress exponent
of 4.22 is somewhat higher than most values reported in
the literature, but most observations have been concerned
with only the secondary or minimum creep-rate. The value
4.22 presented here is believed to reflect the effect
Of a relatively higher equilibrium value of S for higher

stresses.

tI

The last part of equation (3-25a),[C3 In (1 + Ce-m
- C3 In (1 + C)],represents an amount of strain that must
be subtracted from what was predicted by the remaining
terms when creep has taken place with the structure term,
S, below its equilibrium value. One would expect that the
ultimate magnitude of this quantity, C3In (l + C), would
have some physical significance, but the exact nature of
this significance is elusive. Some insight may be gained
by computing the amount of strain the various samples had
undergone when the term C3 In (1 + Ce-mt), hereafter called
the softening term, had decayed to some fraction of its
original value. Figure 6-12 presents the results of these
computations with the C value equal to 62.5. It appears

that the softening term decays to 0.5 of its original

value at about 2.2 per cent strain. By the time the

96

softening term has reached 0.1 of its original value, a
definite bilinear relationship has developed.

Weertman (1963) has accounted for the difference
in creep-rates of polycrystalline ice and single-crystal
ice oriented for easy glide by the fact that the resolved
shear stress on the basal plane of any grain in the poly-
crystalline mass is less than the applied stress. The
angle 0 was defined by Weertman as the angle that the
normal to the basal plane makes with the direction of com-
pression. Then the shear strain along the basal plane
must be multiplied by cos 0 sin 0 to obtain the resolved
strain in the direction of the applied force. Steinemann
(1954) found that ice single crystals soften by the time
they have reached 20 per cent strain along their basal
planes. The last crystals to soften in a polycrystalline
mass subjected to uniaxial compression will be those with
orientation such that 0 equals 45 degrees. Thus, since
sin 45° cos 45° equals 0.5, approximately 10 per cent
axial strain must occur in polycrystalline ice before all
crystals have opportunity to soften. This is believed to
account for the bilinear relationship evident in Figure
6-12.

For high stresses the equilibrium value of S is
relatively large. As creep proceeds, the instantaneous
value of S increases, leading to an accelerated creep-

rate. This in turn leads to a larger value of S, and so

97

forth; thus large strains are required for S to reach its
equilibrium value. At lower stresses the equilibrium
value of S is low, and should be reached at low strains,
however, 10 per cent axial strain is required to give all
crystals Opportunity to soften. Consideration of
Figure 6-12 indicates that for stresses below about 1.7
MNm"2 (250 psi) S is strain controlled, while for stresses
above this value, S is time controlled. This is the reason
that the best fit line for the C3 versus stress plot of
Figure 6-11 is shown broken.

Based on the preceeding results, equation (3-25)
can be expressed as

8.28 x 10'504'85t -4 4.22

s = 0.581e' + 1.89 x 10 a t

0’

5.83 x 10 t

+ C3In (l + 62.5e' ). (6-4)

where C3 equals 2.65 per cent for stresses up to 1.7 MNm-2

(250 psi) and 3.940 - 3.92 per cent for stresses of

2 to 2.74 mm"2

1.70 MNm“ (400 psi).

According to the assumptions inherent in equations
(3-25), if the load on a sample were changed abruptly at
some time after the primary creep term has decayed to an
insignificant amount, the creep-rate would change imme-
diately and then simply decelerate or accelerate to some

new constant value depending on whether S is initially

98

greater or less than the equilibrium value for the second
load. Figure 6-13, a creep curve for sample 10B, indi-

cates that this is the case if the load is abruptly

increased. The loading was increased from 1.38 MNm--2

(200 psi) to 2.07 MNm.2 (300 psi). Figure 6-14, a creep

curve for sample 113, shows an initial deceleration with

reacceleration when the load is abruptly reduced. The

loading was reduced from 2.07 MNm-2

MNm_2 (118 psi). Note that the final creep-rate compares

(300 psi) to 0.695

favorably with that predicted by the appropriate value of

C as shown on Figure 6-10. The final creep-rate for

2!
the second loading phase in Figure 6-13 is 3.51 x 10.3 % s-1

3 -1

compared with C value of 4.0 x 10- % s . The final

2
creep-rate for the second loading phase in Figure 6-14

is 5.10 x 10—5 % 5.1 compared with C2 value of 4.10 x 10-

% s-l. The behavior of sample 11B is believed to result

5

from a decreasing 8 value after the load is reduced. A
rather large recrystallization rate, corresponding to a
large S value for the larger first load, continues for a
time after the load is reduced, and creates a large number
of fresh crystals within a rather small range of strain.
These fresh crystals must all undergo about the same
strain to soften, and since they are not immediately
available for softening, they cause the creep-rate to

undershoot its equilibrium value.

99

A series of tests, 12B through 15B, performed with
two phases of creep, were tested under an axial stress of

1.71 MNm-2

(250 psi) in order to determine the value of
the recrystallization constant K4'. Figure 6-15 shows
the creep curves for these tests, and Table 3 lists the
results. In this series of tests the load was applied,
removed for varying lengths of time, and then reapplied.
It was assumed that the second loading phase started at
time equal to zero so that equation (3-25) can be applied.
The parameters listed in Table 3 for the second loading
phase were obtained by a least squares best fit, similar
to the previous technique. Included also in Table 3 are
the actual strains at the end of the first loading, along
with the computed value according to equation (3-25)
where parameters were used from Table l. The first phase
loading was too short for reliable curve fitting.

Referring to Figure 6-11, parameter C is constant

2

3

for stresses up to 1.71 MNm- (250 psi), and from equation

(3-28),C2' = 2 C2/C3 - m, where m and C2 are stress

dependent. According to Figures 6-9 and 6-10, the term
C ' C '

2 2
C TC

—5_ 3 II in equation (3-40) should be zero

3 I

since the applied stress was 1.71 MNm-2

(250 psi) for both
loading phases in this series of tests. The subscript I
indicates first phase loading and II indicates second

phase loading. Thus equation (3-40) can be simplified as

(6-5)

where B2, B and D have been defined before. The values of
K4' were computed by this equation, using first phase
results of sample 5B. Results are shown in Table 3. The

consistency of the resulting K4' values is satisfactory.

The average value of K4' is taken as 3.322 x 10-4 5-1.

The values of parameter Kz' were obtained from

equation (3-41) using this K4' value, and the equilibrium

values of Ss were obtained from the equation C2 = Ss KZ'

or from equation (3-17) for this K ' value. Table 4

2
lists the values of K ', and Ss, along with values of

2
K ', Cl', Cz', C3' and K2' So which were computed according
to the formulas indicated in the corresponding columns in
Table 4.
Figure 6-16 displays a plot of K2' versus stress
on a logarithmic scale. Using an hyperbolic sine function,

the equation representing the best fit curve was obtained

as

, _ -4 . __g;__ _
K2 — 3.65 x 10 Sln h 0.583' (6 6)

This agrees with the original formulation as in equation

2

(3-4).- In this manner, 00 = 0.583 MNm- (84.6 psi).

101

From the expression 3L = [32%, where k = 1.38 x 10-16
o

ergs/mole °C (Boltzmann's constant), T = 2.687°K (-4.45°C),
and 00 = 84.6 psi, one can calculate the flow volume qAI
equals to 2.71 x 104 cubic angstroms for this series of
tests. The size of flow unit indicates that boundaries of
ice grains may be involved in the deformation process.

From thermodynamics F equals H - TAS, and

equation (3-3) can be rewritten as

E = /2 15111-exp (- %%) sin h (KZ.Q§&.O)

h 3 2kT
or
° — E1 - BE. ' £L _
8 — /2 11 exp ( RT) Sln h do. (6 7)

Equation (6-7) was originally shown to be equivalent to

equation (3-5), i.e.,‘s2 = SKZ'. Now with the result

Kz' = 3.65 x 10.4 sin h 0—%83’ it is apparent that the term

term /2'%$-exp (-.%%) in equation (6-7) must be the
multiplication of‘S and the constant 3.65 x 10-4. Since
parameters k, T, h and R are known values, then taking

the S value at equilibrium from Table 4, one can calculate
the activation energy AF. AF was obtained with a range
from 20.7 Kcal/mole for test sample 13 to 20.0 Kcal/mole
for test sample 9B. These values agree favorably with the
values obtained by Jellinek and Brill (16.1 Kcal/mole),

Nakaya (18.7 Kcal/mole), Landauer (14 to 27 Kcal/mole) and

102

Dillon and Andersland (21 to 25 Kcal/mole). The agreement
is somewhat surprising because these earlier investigators
considered the secondary portions of creep. In addition,
the numerical values of Ss shown in Table 4 confirm the
earlier hypothesis that for higher stress, the equilibrium
value of S is relatively higher. The values of Ss were
obtained from 0.042 for test sample 1B to 0.8296 for test
sample 9B. Figure 6-17, a plot of Ss versus stress,

shows that the value of Ss appears to be approaching some
constant Value. This indicates that there is a limiting
value for S. Based on the original assumption O i 8‘: 1,
this limit should be 1.

Table 5 lists the value of S computed from test
sample lB through 9B according to equation (3-16) for the
times of 3000 seconds, 6000 seconds and 9000 seconds.
Parameter values for these calculations were taken from
Tables 1 and 4. It can be seen from Table 5 that high
stress leads to a higher value of S, and as time becomes
large, the S value approaches 85. This result confirms
the earlier prediction.

Figure 6-18 shows plots of K 'S K '50 and C '

2 s' 2 2
versus stress on logarithmic scales. The values of KZ'Ss
are actually equivalent to the value of C2, where C2 is
the final creep-rate when S reaches equilibrium. Since
the samples were stressed in the mold in preparation and

in preloading, recovery to a stress free condition prior

103

to testing is doubtful. Therefore, K '80 must be the rate

2
created by some initial residual S, which existed in the
samples. It appears for the stresses and temperature
investigated that any creep-rate, after the primary creep
due to grain boundary effects, will fall into the band
between lines K2'Ss and KZ'SO.

The exact physical significance of C2' is elusive.
However, it is interesting to note that the plot of C2.
versus stress on logarithmic scales is within the band

made by K2'Ss and KZ'So lines and nearly parallel to them.

Low Stress Level

 

The stress and strain-rate relationship for poly-
crystalline ice creeping steadily under low stress has not
been firmly established. The linear relationship between
stress and strain-rate in earlier investigations by
Butkovich and Landauer (1960) and by Bromer and Kingery
(1966) is questionable. Recently, Mellor and Testa (1969)
performing uniaxial compressive creep tests on fine-grained
polycrystalline ice indicated that the secondary strain-
rate is pr0portional to stress to the 1.8 power. This
result agrees well with this present study. However, the
experimental results and theoretical explanation presented
here give a more reasonable approach for explaining the

mechanism of flow at low stress.

104

2 (23

A creep curve for a stress of 0.1586 MNm-
psi) is shown on Figure 6-19. Creep was allowed to proceed
until an apparently steady strain-rate developed. The
duration of the test was about 500 hours, and it appears
from the curve that the secondary creep rate was in progress
at least 300 hours. The steady strain-rate in this curve
was 2.18 x 10 sec .

Comparing the present result with Figure 4 presented
in the paper by Bromer and Kingery (1966), one finds a

1 2

value of 103/T for this study of 3.73°K- , and n(GS)- for

16

this study is 1.08 x 10 poise/cmz, where the grain size

GS was taken as an average value 1.5 mm, the viscosity n

was computed 2.43 x 1014

poise. This point, 11(GS)-2 versus
103/T, is located far off the line on their Figure 4.
Therefore, their activation energy of 12 Kcal/mole does not
agree with this study. This is true since the previous

results found from series B tests were between 20 Kcal/mole

and 21 Kcal/mole.

AF

On the basis of the expression é = A exp (- fig),

where A is a constant, using an activation energy of

20 Kcal/mole, strain-rate of 2.18 x 10-9 sec"1 and teme
perature of -4.45°C, one can solve for A. One then obtains
a strain-rate of 3.04 x 10.9 sec.1 corresponding to a
temperature of -2.06°C. This result agrees well with the
previous results obtained by Mellor and Testa (1969).

Figure 6-20 (after Mellor and Testa) showing the plot of

105

axial strain-rate versus axial stress at a temperature
-2.06°C is illustrated for comparison.

Even though this experimental data agrees with
the former results, it seems not to satiSfy the present
theory described in the preceeding chapter. Since steady-
state tertiary creep was not developed, application to the
flow law, equation (3-25), is invalid. From earlier
experimental observations, nearly 10 per cent of total
strain is needed before tertiary steady-state creep is
reached. In reviewing the current experiment, for 500
hours test duration, 1.08 per cent of total strain is
still far below that necessary for full tertiary creep.

9 sec-1) approximately

At the current strain-rate (2.18 x 10-
4 x 107 seconds (463 days) additional time would be needed
to reach 10 per cent total strain. Since the test durations
are inordinately long for lower stress levels, this might

be the reason that most experimenters have merely investi-
gated the secondary stage of creep. In addition, the
reliability of the test equipment and sample deterioration
will become serious problems.

Thin-Walled, Hollow, Cylindrical
Ice SampIes

 

 

 

Constant Axial Stress
Qgeep Test

Theoretically, creep tests under constant axial

stress on thin-walled, hollow, cylindrical polycrystalline

106

ice samples should be the same as that for solid,
cylindrical ice samples. Results of test sample 1H,

2 (209 psi), are

subjected to an axial load of 1.441 MNm-
shown on Figure 6-21. The usual three stages of creep,
primary, secondary, and tertiary are evident. Constant
strain-rate developing after nearly 10 per cent strain
agrees very well with the former observations of solid,
cylindrical sample tests. Best fit parameters of equation
(3-25) were obtained and are listed in Table 6. Also
listed in Table 6 are parameters computed from series B

2

tests for a stress of 1.441 MNm- (209 psi). Experimental

errors are believed to be responsible for the variations.

Constant Tensile Creep Test

Figure 6-22 shows a creep curve for a constant
tensile creep test. The tensile stress 0.510 MNm-2
(74 psi) was computed from the internal pressure of 0.081

MNm.-2 (11.8 psi) by using equation (3-62). Even though the

test duration was 4.3 x 104 minutes, the tertiary stage of
creep had not been fully reached. The total strain at the
end of test was only 8.4 per cent. Nearly 10 per cent
strain is necessary to develop fully tertiary creep
according to the former results. The erratic sample
behavior apparent between about 0.5 per cent strain and
2.5 per cent strain is believed to be due to lack of
accurate control of the stress applied to the sample.

The control of this stress was restored during the remainder

of the test.

107

The final strain-rate was 1.15 x 10-7 sec-l. Con-

verting this present result to correspond to -2.06°C,
as before, one obtains a strain-rate of 1.55 x 10-7 sec-l.
This result agrees very well with results by Mellor and
TeSta (1969) (see Figure 6-20). This test may be con-
sidered to have been run at "low" stress. A stress of 85
psi was defined as "low" by Dillon and Andersland (1967).
These results indicate that creep on ice in
tension is the same as for that in compression. Krauz
(1962) performing creep tests on columnar ice beams of
different thicknesses reached a similar conclusion.

Constant Biaxial Stress
Creep_Test

 

 

A combination of axial load and external hydro-
static pressure was applied to test sample 3H. ‘Figure
6-23 presents the creep curves for axial deformation by
axial loading andlateral deformation by hydrostatic
pressure. Both curves appear to approach a steady strain-
rate at full tertiary stage. Assuming volume change is
zero, (Goughnour, 1967) one obtains the octahedral shear

strain by applying equation (3-47)

 

2:

. = % /I€l-€2)2 + (€2-E3) + (83-: 2

Y ) (3-47)

oct 1

where s = — (61+83).

108

The octahedral shear stress due to the principal stresses

was computed according to equation (3-46)

 

2 2

T = % /Q01-02)2 + (0 -0 + (03-01) . (3-46)

oct. 2 3)

Toct. was found equal to 0.501 MNm.2 (72.7 psi).

Figure 6-24 shows a plot of time against octa-
hedral shear strain. This result is as expected showing
three stages of creep. In addition, a steady octahedral
shear strain-rate was developed during the tertiary stage
of creep. After applying the creep curve data to equation
(3-50), the best fit parameters were obtained. Table 7
lists these results. Also listed in Table 7 are the
parameters converted from the result of series B tests
into the octahedral sense for the octahedral shear stress

of 0.501 MNm-2

(72.7 psi) according to the method described
in Chapter III. The relationships for the parameters and

stress in the octahedral sense for series B tests are:

_ -3 4.85 _
noct. — 3.15 x 10 (Toot.) (6 8)

_ -3 4.05 _
moot. - 1.22 x 10 (Toct.) , and (6 9)

_ -3 4.22 _
CZoct. — 6.36 x 10 (Toct.) . (6 10)

109

As can be seen from Table 7, the parameter values
obtained from test sample 3H are relatively higher than
those of series B tests, but the difference is small and
believed to be within permissible errors.

The plots of equations (6-8), (6-9), and (6-10)
are shown on Figures 6-25, 6-26 and 6-27. The circled
point on each figure is the result from test sample 3H.
Since the original relationships between n, m and C2, and
the stress 0 from the series B tests are the average
from the figures, the discrepency between results of test
sample 3H and series B tests is expected.

For the three applied principal stresses

01 = 1.324 MNm'2 (192 psi)

02 = 0.0966 MNm"2 (14 psi)
-2 .

03 = 0.683 MNm (100 p31),

the corresponding stress deviators, by equation (3-45),

were obtained as

01' = 0.6207 MNm-2 (90 psi)
. _ -2 .
02 ——0.607 MNm (- 88 p81)
03' = - 0.0138 MNm-2 (- 2.0 psi),

110

The strain-rate will be taken at the nearly constant
strain-rate region in the creep curves where the structure
of the ice appeared to be at equilibrium and the flow of
ice was nearly steady state. Arbitrarily taking a time of

1100 minutes, the strain-rates were respectively

8 = 3.42 x 10-6 sec-1

1.42 x 10.6 sec-l.

0')
ll

AsSuming no volume change in ice during creep, there

exists an expression

8 + s + s = 0 (6-11)

8 + e + e = 0 ' (6-12)

Thus 82 is equal to -4.84 x 10-6 sec-l.

Figure 6-28 presents the resulting vectors for both
stress deviator and strain-rate components on the same
three coordinates. As can be seen from the figure the two
resulting vectors are not parallel. However, the cross
angle is small. The deviation between experimental results
and the theoretical assumption is attributed to impurity
of ice material (nonhomogeneous and anisotrOpic) or experi-

mental errors.

111

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Axial Strain %

Axial Strain %

16

14

12

10

 

 

30

113

(a) Showing Three Loading Cycles

Rest 7.344 x 104 5

Rest 7.764 x 104 s

l 1 1 1 l 1 l l J l I l L

60 90

Time min.

(b) Loading Cycles Superimposed

3rd Loading

S
lst Loading ample 6A

T = -4.45°C

o‘ = 1.715 mun'z

2nd Loading

1 J l 1 1 I 1 1 J I l l l

l

120 150 180 210 240 270 '300 330 360 '390 '420 450

(249 psi)

 

10

20 3O 4O 50 60 70 80 90 100 110 120 130 140

Time min.

Figure 6-3.--Creep Curve for Sample 6A.

114

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115

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116

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117

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118

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Residuals %

Axial Strain %

22

20

18

16

14

12

10

119

 

 

10

20

l
30

 

I
40

Sample BB
T - -4.45°C

o = 2.38 MNm-2 (350 psi)

I l l l l

50 60 70 80 90
Time min.

Figure 6-5.--Creep Curve for Sample 8B Showing

Correlation with Equation (3-25).

120

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121

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122

 

 

 

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104..
r-
- 1 1 l LIlllllllIIlllllllliil
10
0.5 - 1.0 1.5 2.0 2.5 3.0

Axial Stress MNm"2

Figure 6-8.--Best Fit Values of n from Equation
(3-25) Versus Axial Stress.

123

10

IIIII—I

10

10

r TT—III'

I

 

 

ilIiJIIIIllililIilll

0.5 1.0 1.5 2.0 2.5 3.0
Axial Stress MNm-

Figure 6-9.-Best Fit Values of m frun Equation (3-25)
versus Axial Stress.

124

 

 

 

10.2 C
I.
10"3 .
I.
H I-
I
m I-
N
r-
N
U
10'4 -
P
10-5 1 l l l lllllLlllIlLLJlIllLII
0.5 1.0 1.5 2.0 2.5 3.0

Axial Stress MNm-2

Figure 6-10.--Best Fit Values of C2 from
Equation (3-25) Versus Axial Stress.

125

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Axial Strain %

126

 

 

 

 

 

40 b

30 —

20 7

10 ..

I 0 J l 1
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‘Axial Stress MNm"2

Figure 6-12.--Axial Strain Necessary for Softening
Term to Reach Stated Per Cent of its Original Value as a
Function of Stress.

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130

       

 

 

10'1 _
L-
-2
10 b , _ -4 . 0
: K2 — 3.68 x 10 Sin h 6:353-
01
w i
-N __
M
10'3 _
y-
10-4 L l 1 njlnnllllnilijjilninll
0.5 1.0 1.5 2 2.0 2.5 3.0

Axial Stress MNm—

Figure 6.16.--Values of K2' Versus Axial Stress.

131

 

 

 

l
E
L.
m #-
U)
1o'i
P-
10- L 1 l 1 [111L141L4J1111111111
0.5 1.0 1.5 2.0 2.5 3.0

Axial Stress MNm—2

Figure 6-17.--Plot of Equilibrium Values of S
Versus Axial Stress.

 

10"2 f

r

L

10"3 t

m I

w I-

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M

B” F

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M

10" r

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p.

10"5 :

0.5

132

 

 

Stress.

1.0 1.5 2.0
Axial Stress MNm"2

Figure 6-18.--Computed K2'Ss, Cz' and K

2

I
So Versus

133

.mma mamfimm 00H mafiaamummHUMHom How m>uso mmmHUII.mHIo shaman

musom mafia

omm omv ovw 00¢ 8m omm omN ovN DON om.” ONH om 0v 0

 

J _ . q _ _ a q _ d _ d d

       
 

N.o
m.o
v.0

00m K .
a- macs ma N

lama mmc «uszz mmma.o
Uomvofil u

u
0w

m.o

E-‘b

mod mamsmm

 

 

% urexns IPIXV

 

Axial Strain Rate 5

10

I YI‘TIIl

r

10

vrxl‘

I

10 ,.

10

#11 TT‘Vf'l

10

I fir!!!

 

T

134

= -2.06°C

  
   

Sample 2H Result

Sample 168 Result

I 1 J 1

 

0.1 ‘ 1.0 10 100.
Axial Stress Kgf/cm2

Figure 6-20.--Axial Strain-Rate Versus Axial Stress

(After Mellor and Testa, 1969).

135

CA N mm

.mH mamfimm How 0>HSU mmmHUII.HNIm musmflh

.cfiz mafia
om mv 0v mm on mm ON mu OH m o

 

00m. w
HI

lame moNV m

MI.

      
    
 

_ . q _ . _ . _ . 7 _

OH x mmo.H u w
.52: Hee.H use

come.v- u a
ma mflmsmm

  

 

OH
NH
2.
0H

ma

% urexns terxv

136

OHN mH

.Aumme mmmuo mHstmBV mm mHmEmm sou m>uso mmmuunl.mmlm wusmHm

 

.cHz mEHB
¢H mH NH HH 0H m m n o m v m N H o
1 d _ . - H _ H _ — 4
0mm x . u u
HI N...0H mH H .
lame «as «:32: oam.o u no
Uomv.¢l u .H. L
umma mwmuo wHHmsma
mm mamsmm 1
L

  

          

 

% urexns IPTXV

137

 

.5 398m now 9.50 m8uul.mmlm musmHm

 

.52 22.
«S x Z 3 NH S 3 m n o m e m m H o
— u — q - d fi — d A u d O)<“l°..0.c
q 4 O O O
1 O
a. q 0 O
0
q o
C 0
00m OH x Nv.H Imw a 0
HI 0| I o d
a c
Q
coHumEHomwa Hmuwug
O
O
o Ream code «-52: mmm.o u mo
. .. m
a can 0 ca x ~e.m "Hm o lame vac «-52: memo o 1 Ho
. o isms «may muszz «mm.H u o
Uomfiofil “ E
0 mm «38mm
0

“83.9.50me Hausa

 

0H

NH.

vH

0H.

mH

% uIexns IEIXV

 

Octahedral Shear Strain %

4O

35

3O

 

l 2 3 4 S 6 '7 8 9 10

138

Sample 3H
T = -4.4S°C
T = 0.501 MNm-2 (72.7 psi)
.oct. _4 _1
e = 7.3 x 10 % sec
oct.

 

L l 1 l l l l l l l l L l L ’

2

11 12 13 14 x 10

Time Min.

Figure 6-24.--Octahedra1 Shear Strain Versus Time for

Sample 3H.

139

   

 

 

10"1 f
-2
10 L
h -3 4.85
_ oct. - 3.15 x 10 (Toot.)
,
P
m ..
J
o
J) .
10”3 -
' Sample 3H Result
-4
10 l 11 #1111 1 1 JlLlllJ
0.1 , 1 10

Octahedral Shear Stress MNm.-2

Figure 6-25.--Plot of “oat Versus Octahedral Shear
Stress for Samples 1B through 9B Compared with Sample 3H.

140

 

 

   

 

10-2 r
r
_ -3 4.05
OCt. — 1.22 x 10 (Toct.)
10'3 -
F
m b
43
U
0 I-
5
Sample 3H Result
10"4 -
P
r
10-5 1 lJJLlLl 1 1 11111LJ
0.1 1.0 10

Octahedral Shear Stress MNm—2

Figure 2-26.--Plot of moct Versus Octahedral Shear

Stress for Samples 1B through 98 Compared with Sample 3H.

10

10

C2 oct. % S

10

TTUIj

rrlli1

 

Figure 6-27.-Plot of C

141

_ -3 4.22
- 6.36 x 10 (root)

C2 oct.

 

3H Result

11L141l 1 1 141111]

1.0 p 10
Octahedral Shear Stress MNm.-2

2 col: Versus Octahedral Shear Stress for

Samples 1B through 9B Cmpared with Sanple 3H.

142'

1 k

.mm mHmEmm How mummlsHmuum can

HoumH>ma mmwnum mo muouom> mo coHuwucmmmumwm UHpmfimnom11.m~1m musmHm

 

 

 

EXT
m . o H H H IIIIIIIIIIIIIIIII Inllllllh H H H H IIIIIIII MN
: u
__ _ [8|
_ _ _
_ _ _
_ _ _ [Dbl
__ _
_ m _
__ w IIIIII _L$MJ.
I \ _ \J
_ _ \\ _ _ .\ _
_ _ x... -u- .ewmmvé- "
_ _ _ _ _
_ _ _ _ .
__ _ _ YLLovu _
__ _ _ . 09.
__ _ _ n u %v 60
. _ . _ 31.8.. _ r .w .
__ n u . _ +A..r my
_ _ _ _ _ a r...
__ _ _ NITTONI _ m
__ _ _ . _ 9
_. . _ _ _ é
__ . \r---- _-.2.. o
_ _ _ \ _ f
1.3% in x... .... _. X m .1 .
Ell- .l' —III a I d Ill-'Ilqllllq1111l' V
is as om am a» .8. an. 8 on 8 2
)w s
. \ 0’
a
+ \
Q? A? % same
\ com
9 AH: my .mo

 

143

OH x mm.H m.mm OO.m OH x mO.H OH x hm.w wem.O oov Owh.~ mm

 

 

m1 m1 m1
mIOH x mm.H m.mh ow.v mIOH x mu.h mIOH x -.O «Hm.o omm OOMwN mm
¢IOH x hm.h O.mb va.v mIOH x O~.¢ mIOH x mm.m vmm.o com OOO.~ . ms
«IOH x hm.m h.mm OO.> mIOH x Oh.¢ m1OH x h¢.v oev.o oom OOO.~ mm
«IOH x N>.¢ m.vm mm.m m1OH x mm.H mIOH x mo.~ mmm.o omm OHn.H mm
«IOH x NH.m O.mm mm.~ mIOH x mh.H mIOH x mO.m ovm.o omm OHh.H me
enoH x Om.~ N.S¢H vo.~ eIOH x -.h vIOH x Oh.m MN0.0 OON mom.H mm
mIOH x em.w ¢.mo mh.~ vIOH x w~.m vIOH x HH.H omm.o omH omo.H mm
mIOH x H~.H v.mv Hm.~ mIOH x NH.v mIOH x OH.H OO0.0 OOH Omo.O mH
Hum M Hum m Hum Hm Hmm WIEZZ .oz
2 o o o a 0 39:8

mmmuum HMHX¢

 

.Uomvofil " B
.mumme m mmHme on uHm ommm How owumsHm>m Ammlmv cOHumswm mo mumqumumm11.H mqm<a

144

 

 

 

 

muoH ma.~ m.~o mm.m muoa mm.H mnoa em.a Hmm.o cow oeu.~ mm
muoH me.a m.~m mm.e muoH ~m.e muoH H¢.m Ham.o omm omm.~ mm
euoa mm.e m.~m ofl.m muoH m~.v muoa em.m Hmm.o com ooo.~ me
euoa «v.5 m.~e m~.m m10H me.e «IOH mo.~ Hmm.o oom .oeo.~ mm
eucH em.¢ m.~o e~.m mnoa mo.a m10H mo.a Hmm.o omm oan.a mm
«nod ~m.m m.~e nm.~ muoa me.H muoH mm.m Hmm.o omm 0H5.H me
euoa ~e.~ m.mm mm.~ enoa mm.e euoa mm.e Hmm.o com men.» mm
muoa om.e m.~e em.~ e-OH om.~ euoa Ho.H Hmm.o oma omo.a mm
muoH m~.H m.~w ~m.~ muoa ~H.¢ muoH ee.H Hmm.o ooa ome.o ma
o mw H- aw “mm «.222 mwmmmm
s o o a o mmmuum Hmsxe
.emxflm mmsam> u can Ho cuss

.mumwa m moHHmm ou uHm umwm MOM omumsHm>m Ammlmv QOHumswm mo muwuofimumm11.~ mamas

145

v

m IOmemm.m u . M mommm>¢

 

 

 

 

H1 v
vIOH x Nmm.m vIOH x om.o m.mm mm.m MIOH x mm.H MIOH x mo.m mmm.o oomh MH.v mm.v comb mm
vIOH x whm.m vnOH x hm.w O.mo mv.H mIOH x Nh.H MIOH x vo.¢ mov.o oovm mH.v vm.v comb mvH
vIOH x vhv.m vIOH x mm.h m.OH vo.~ MIOH x om.H mIOH x vm.m mom.o ooom Om.g om.v OVHS mm
vIOH x Omm.m «10H x mm.w mm.m mv.v MIOH x mm.m NIOH x ov.H mmm.o oowH mH.¢ mm.v comb mN
m m a ma w w w m
Hlv H1 E 0 mo H1 NU Hm Ho m cwmuum chuum made um.
. M wade Omusmeoo Hmouom . .
cmusmfiou mmmam OQHUMOA ocoomm ummm mmmnm OCHGMOA umHHh

 

v

.Asmm ommv «-52: He.a

mo mmmuum paw oomv.v1 um mumme HH¢ .. M ucmumcoo coHHMNHHHmumhnomm mo wsHm> wcHEHmswa ou mamas mo muHSmmmI1.m mqm,

 

 

146

 

 

¢IOH x O.m mIOH mmm.v MIOH x mum.H mIOH x hmm.mu OHH.O mmmm.o NuoH x mo.N vh.~ mm
vIOH x m.m vIOH mHN.H MIOH x mmm.H MIOH x SOH.NI mHm.O thw.o NIOH x HO.H mm.~ mm
vIOH x 0.0 mIOH Nvm.b vIOH x wo>.m MIOH x mom.H1 OON.O mHm0.0 MIOH x mm.m oo.~ mu
«10H x m.m mIOH mmw.H vIOH x mmmro vIOH x th.m1 ONH.O wmmm.o mnoH x Hm.h OO.N mm
«10H x m.m mIOH mnm.m vIOH x mnm.m vIOH x mm~.m1 mmm.o omnm.o maoH x mm.m Hh.H mm
vIOH x ¢.~ mIOH mom.m «IOH x mvm.o m10H x nNO.H1 mvm.o nmmm.o mIOH x OO.m Hh.H mo
«IOH x mv.H mIOH me.m vIOH x mHN.v v10H x ovm.hu omv.o vmmv.o mIOH x vm.H mom.H mm
mIOH x b.m wuoH mom.m mIOH x vov.m «IOH x mo~.v1 mmm.o nHmH.o mIOH x mH.H mO.H mm
OIOH x m.h h10H th.H mIOH x NNh.H @10H x www.mn 0mm.o omeo.o vIOH x mm.o omm.o mH
AHmumO N Ammnmv AOHIMO Anmlmv N m AHvumv 52: .oz
coHumsvm A E: . UYIII u :oHumsvm coHumsUm coHumswm . M\ U u codumswm N1
N , . mmwuum onEMm
mm m m 1m w m w
07m Hm Hm I T 7 Have
W. Vm - U . U a O .MVH mm .Nx
.momme 0 muss o Ho 0 N m . N . H . v N .11
um m m w m. M can . U . U . U . M .mm .. M NO mwsHm>Il.v mqmda

147

TABLE 5.--Value of S as a Function of Stress and Time.

 

 

Axial

Sample Stress S - S S
No. MNm-2 T=3000 sec. T=6000 sec. T=9000 sec.
1B 0.686 0.00809 0.00812 0.00814
2B 1.03 0.0319 0.0326 0.0334
3B 1.365 0.0961 0.10389 0.1211
43 1.71 0.1029 0.1831 0.3568
SE 1.71 0.1066 0.1939 0.3583
6B 2.06 0.1381 0.3645 0.5561
73 2.06 0.1581 0.3930 0.6019
SB 2.38 0.4710 0.7685 0.7740
93 2.74 0.7177 0.8291 0.8296

 

148

 

 

momuo HMde mauve

 

«IOH x N.N m.mm mm.“ m..OH x OO.H vIOH x O.m Hmm.o va.H m moHuom
¢IOH x Onm.H hm.Om bm¢.> MIOH x omO.H. MIOH x VH.H bmv.o va.H mmwuo Hme< mH
w m w m w 522 .
H1 H1 H1 N1 9 oz
E 0 m0 NU : Ho mmouum m as umma mHmEmm

 

.mH mflmsmm op yum ummm you emumaam>m Ammumv coaumsem «0

mumumfioummll.o mamas

149

 

OH x O¢.h «.mm mh.m OH x ¢¢.m IOH N MOH.H NN0.0 Hom.O mmwuum

 

m1 «1 w muMHumoummm
How mummu
m meumm
«IOH x m~.H me.om me.m euoa x om.h «nos x ~m.H oe.H Hom.o mm
m o 00 m E
H- u. o a H- m Hum w ~._zz moz
. . . . . mmmuum ma saw
#008 “.00 MO “.00 1H0 U00: “00 Ho Hmmgm goo

 

.mgHSmmm m mmHHmm
suH3 oouwmfiou mm mHmEmm op uHm ummm MOM omumus>m mnmumamumm Hmuomsmu0011.h mqmda

CHAPTER VII

CONCLUSIONS

The conclusions summarized below are based on the
observed results of experiments conducted on samples of
polycrystalline ice at a temperature of -4.45°C and the
theoretical analysis presented in this study.

1. Equation (3-25) was derived by a hypothesis
which describes the creep behavior of polycrystalline ice
in terms of the mobilization of grain boundary effects,
softening of individual crystals due to accumulated strain
on their basal planes, and recrystallization. It appears
to account for this behavior through primary, secondary
and tertiary stages of creep for the range of stresses and
temperature studied. The equation predicts a creep-rate
that approaches some constant value when grain boundary
effects are fully developed and as the quantity S approaches
some equilibrium value, where S is defined as the decimal
per cent of soft crystals in the polycrystalline mass.
Experimental results from uniaxial creep tests and multi-
axial creep tests for high stress levels appear to fit this

form of equation for strains up to at least 25 per cent.

150

 

 

151

2. It is observed from experimental results that
approximately 10 per cent of total strain is necessary
to reach the tertiary steady state creep. In other words,
approximately 10 per cent axial strain must occur in poly-
crystalline ice before all crystals have opportunity to
soften. This is in agreement with results obtained by
Steinemann (1954).

3. A value of about 20 Kcal/mole for activation
energy was found for sample tested at a temperature
-4.45°C. This value agrees favorably with results by
previous experimenters.

4. The results from creep tests under low stress
seem not to satisfy the theoretical creep equation (3-25).
Since steady-state tertiary creep was not developed for
these tests, application to the equation (3-25) is not
valid. However, the secondary creep-rate from these tests
agrees well with previous results by Mellor and Testa
(1969). The lack of a tertiary creep stage may be due to
the fact that these tests were not continued to a total
strain of 10 per cent.

5. The validity of the assumption by Nye (1953)
that for polycrystalline ice the principal strain rates
are parallel and proportional to the corresponding princi-
pal stress deviators was verified by the tests on thin-

walled, hollow, cylindrical ice samples. The results were

 

“A.“ -

152

satisfactory. Discrepancies are attributed to impurity
of ice material (nonhomogeneous and anisotropic) or

experimental errors.

BIBLIOGRAPHY

153

BIBLIOGRAPHY

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Bromer,

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R. L. "A Model for Boundary Diffusion Controlled

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Atoms Round a Slow Dislocation." Proc. Royal Soc.
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H. B., and Andersland, O. B. "Deformation Rates

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154

 

155

Dorn, J. E. "Some Fundamental Experiments on High Tempera-
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Dorsey, N. E. Properties of Ordinary Water-Substance.
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Friedel, J. "Dislocation Interactions and Internal Strains
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Garofalo, F. Fundamentals_of Creep and Creep-Rupture in
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Glasstone, 8.; Laidler, K. J.; and Eyring, H. The Theor
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Glen, J. W. "The Creep of Polycrystalline Ice." Proc.
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Glen, J. W., and Perutz, M. F. "The Growth and Deformation
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Gold, L. W. "Some Observations on the Dependence of
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Gold, L. W. "Deformation Mechanisms in Ice." Chapter 2,
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Goughnour, R. R. "The Soil-Ice System and the Shear
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Goughnour, R. R., and Andersland, O. B. "Mechanical
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Griggs, D. T., and Coles, N. E. "Creep of Single Crystals
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Halvorsen, L. K. "Determination of the Modulus of
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156

Hawkes, L. "Some Notes on the Structure and Flow of Ice."
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Herrin, M., and Jones, G. E. "The Behavior of Bituminous
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Herring, C. "Diffusional Viscosity of a Polycrystalline

Solid." J. of Applied Physics, 21, 1950, pp. 437-
445.

Higashi, A., and Sakai, N. "Movement of Small Angle
Boundary of Ice Crystal." J. Faculty of Science,
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Itagaki, K. "Self-Diffusion in Ice Single Crystals."
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Jellinek, H. H. G. "Plastic Deformation of Thick-Walled
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Jellinek, H. H. G., and Brill, R. "Viscoelastic Properties
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Kamb, W. B. "The Glide Direction in Ice." J. of
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Kanter, J. J. "The Problem of the Temperature Coefficient
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Kauzmann, W. "Flow of Solid Metals from the Standpoint of
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Krausz, A. S. "The Creep of Ice in Bending." Canadian J.
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Leonards, G. A., and Andersland, O. B. "The Clay Water
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157

Levi, L.; DeAchaval, E. M.; and Suraski, E. "Experimental
Study of Non-basal Dislocations in Ice Crystals."
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Lonsdale, K. "The Specific Gravity of Ice." J. of
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Main, J. F. "Note on Some EXperiments on the Viscosity of
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491-501.

 

McConnel, J. C. "On the Plasticity of an Ice Crystal."
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McConnel, J. C., and Kidd, D. A. "On the Plasticity of
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44, 1888, pp. 331-367.

 

Mellor, M., and Testa, R. "Creep of Ice Under Low Stress."
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Muguruma, Jo, and Higashi, A- "Observation of Etch
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Nabarro, F. R. N. "Deformation of Crystals by the Motion
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the Strength of Solids, the Physical Society,
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Nakaya, U. "Mechanical PrOperties of Single Crystals of
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Nakaya, U. "Viscoelastic Properties of Snow and Ice in
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Nye, J. F. "The Flow of Ice from Measurements in Glacier
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fraufirn Borehole Experiment." Proc. Roy. Soc. of
London, Ser. A, 219, 1953, pp. 477H489.

 

Pounder, E. R. The Physics of Ice. Pergamon Press,
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158

Rigsby, G. P. "Crystal Orientation in Glacier and in
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3, 1960, pp. 589-596.

Rigsby, G. P. "Effect of Hydrostatic Pressure on Shear
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Glaciology, 3, 1958, pp. 273-278.

 

 

Shoumsky, P. A. "The Mechanism of Ice Straining and Its
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Shumskii, P. A. Principles of Structural Glaciology.
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Steinemann, S. "Results of Preliminary Experiments on
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Weertman, J. "The Eshelby-Schoeck Viscous Dislocation
Damping Mechanism Applied to the Steady-State
Creep of Ice." Ice and Snow, Properties, Processes
apd.App1ications, ed. by W. D. Kingery, MIT Press,
Cambridge, Mass., 1963, pp. 28-33.

APPENDIX

DATA

159

160

 

 

 

SAMPLE 1A SAMPLE 2A
T = -4.45°C T = -4.45°C

Y1 = 0.902 gm/cc Yi = 0.905 gm/cc

o = 350 psi . V..,o = 350 psi .......

Time Deflection Time Deflection

(min.) (in.) (min.),, .(in-)..
0 0.0076 0 0.0076
1 0.0126 1 0.0122
3 0.0173 3 0.0182
5 0.0206 5 0.0224
10 0.0274 10 0.0337
15 0.0344 15 0.0412
20 0.0418 20 0.0523
25 0.0508 25 0.0660
30 0.0620 30 0.0836
35 0.0760 35 0.1054
40 0.0938 40 0.1326
45 0.1162 45 0.1638
50 0.1439 50 0.1984
55 0.1766 55 0.2369
60 0.2138 60 0.2782
65 0.2550 65 0.3228
70 0.2982 70 0.3715
75 0.3441 71 0.3822

Load Removed

 

Load Removed

 

 

SAMPLE 3A
T = -4.45°C
71 = 0.904 gm/cc
o = 350 psi
Time Deflection
(min.) (in.)
0 0.0080
1 0.0134
3 0.0170
5 0.0196
10 0.0258
15 0.0321
20 0.0387.
25 0.0465
30 0.0560
35 0.0673
40 0.0816
45 0.0993
50 0.1214
55 0.1477
60 0.1793
65 0.2160
70 0.2564
75 0.2994
80 0.3440

Load Removed

161

 

 

SAMPLE 4A
T = -4.45°C
Yi = 0.905 gm/cc
o = 350 psi.

Time Deflection
(min.) (in.)

0 0.0104

1 0.0166

3 0.0213

5 0.0246
10 0.0317
15 0.0386
20 0.0459
25 0.0542
30 0.0640
35 0.0758
40 0.0898
45 0.1063
50 0.1262
55 0.1494
60 0.1759
65 0.2059
70 0.2390
75 0.2752
80 0.3135
85 0.3526

Load Removed

 

 

 

 

 

 

SAMPLE 5A
= -4.459C
y. = 0.903 gm/cc
= 350 psi
Time Deflection
(min.) (in.)
0 0.0096
1 0.0157
3 0.0202
5 0.0236
10 0.0314
15 0.0390
20 0.0472
25 0.0568
30 0.0681
35 0.0821
40 0.0990
45 0.1196
50 0.1446
55 0.1734
60 0.2060
65 0.2420
70 0.2908
75 0.3210
80 0.3612
Load Removed
SAMPLE 6A
= -4.45°C
y. = 0.900 gm/cc
= 249 psi
Time Deflection
. (min.) (in.)
0 0.0070
1 0.0106
3 0.0135
5 0.0155
10 0.0194
20 0.0259
30 0.0313
40 0.0362
50 0.0410
60 0.0460
70 0.0510
80 0.0566
90 0.0626

 

Time Deflection
(min.) (in.)
100 0.0696
110 0.0775
120 0.0864
130 0.0964
140 ' 0.1077
Load Removed
Rest 7.764 x 104 seconds
0 0.1088
1 0.1112
3 0.1140
5 0.1156
10 0.1190
20 0.0236
30 0.1282
40 0.0322
50 0.1364
60 0.1410
70 0.1462
80 0.1520
90 0.1585
100 0.1658
110 0.1745
120 0.1846
130 0.1960
140 0.2008
150 0.2232
153 0.2264

Load Removed

t 7.344 x 104

 

Res seconds
0 0.2228

1 0.2254

3 0.2282

5 0.2299
10 0.2333
20 0.2384
30 0.2424
40 0.2464
50 0.2508
60 0.2554
70 0.2604
80 0.2663
90 0.2734
100 0.2813
110 0.2903
120 0.3003
130 0.3112

Load Removed

SAMPLE lB

 

 

 

T = -4.45°C
Yi = 0.9025 gm/cc
o = 100 psi
Time Deflection
-(min.) (in.)
0 0.0032
1 0.0042
10 0.0052
60 0.0070
180 0.0099
360 0.0111
540 0.0169
720 0.0202
900 0.0234
1080 0.0268
1260 0.0301
1440 0.0333
1620 0.0367
1800 0.0400
1980 0.0434
2160 0.0466
2340 0.0497
2520 0.0530
2700 0.0564
2880 0.0597
3060 0.0631
3240 0.0666
3420 0.0701
3600 0.0738
3780 0.0776
3960 0.0816
4140 0.0856
4320, 0.0898
4500 0.0941
4680 0.0988
4860 0.1035
5040 0.1087
5220 0.1142
5400' 0.1200
5580 0.1258
5760 0.1319
5940 0.1384
6120 0.1450

 

Time Deflection
(min.) (in.)
6300 0.1517
6480 0.1586
6660 0.1650
6840 0.1727
7020 0.1803
7200 0.1888
7380 0.1967
7560 0.2047
7740 0.2128
7920 0.2212
8100 0.2292
8280 0.2328
8460 0.2460
8640 0.2547
8820 0.2630
9000 0.2713
9180 0.2795
9360 0.2876
9540 0.2952
9720 0.3033
9900 0.3110
10080 0.3192
10260 0.3270
10440 0.3348
10620 0.3425
10800 0.3500
10980 0.3570
11160 0.3648
11340 0.3722
11520 0.3797
11700 0.3868
11880 0.3938
12060 0.4003

Load Removed

SAMPLE 23

-4.45°C

= 0.903 gm/cc

 

 

 

= 150 psi
Time Deflection
(min.) (in.)

0 0.0030
1 0.0043
10 0.0064
20 0.0076
40 0.0099
100 0.0156
200 0.0239
300 0.0315
400 0.0391
500 0.0467
600 0.0549
700 0.0637
800 0.0746
900 0.0877
1000 0.1031
1100 0.1217
1200 0.1415
1300 0.1635

 

Time Deflection
(min.) (in.)
1400 0.1878
1500 0.2128
1600 0.2381
1700 0.2638
1800 0.2903
1900 0.3166
2000 0.3421
2100 0.3661
2200 0.3891
2300 0.4090
2400 0.4320
2500 0.4520

Load Removed

 

165

 

 

 

 

SAMPLE 3B
T = -4.45°C
Yi = 0.906 gm/cc
o = 200 psi
Time Deflection Time Deflection
(min.) (in.) (min.) (in.)
0 0.0050 340 0.1336
0.0076 360 0.1474
5 0.0104 380 0.1624
10 0.0128 400 0.1780
20 0.0166 420 0.1942
40 0.0226 440 0.2116
60 0.0279 ' 460 0.2294
80 0.0328 480 0.2466
100 0.0378 500 0.2642
120 0.0426 520 0.2812
140 0.0479 540 0.2976
160 0.0530 560 0.3141
180 0.0586 580 ' 0.3306
200 0.0651 600 0.3469
220 0.0722 620 0.3629
240 0.0802 640 0.3789
260 0.0886 660 0.3945
280 0.0978 680 0.4098
300 0.1088 688 0.4158

320 0.1206 Load Removed

166

 

 

 

SAMPLE 4B SAMPLE SB

T = -4.45°C T = -4.45°C
Yi = 0.903 gm/cc Y1 = 0.903 gm/cc

g = 250 psi 0 = 250 psi

Time Deflection Time Deflection
(min.) (in.) (min.) (in.)
0 0.0072 0 0.0060
1 0.0121 1 0.010

5 0.0172 5 0.0145
10 0.0218 10 0.0180
20 0.0290 20 0.0240
30 0.0352 30 0.0290
40 0.0412 40 0.0340
50 0.0453 50 0.0390
60 0.0515 60 0.0440
70 0.0605 70 0.0496
80 0.0684 80 0.0559
90 0.0777 90 0.0627
100 0.0880 100 0.0704
110 0.0990 110 0.0794
120 0.1119 120 0.0892
130 0.1263 130 0.1001
140 0.1418 140 0.1124
150 0.1585 150 0.1265
160 0.1763 160 0.1419
170 0.1948 170 0.1585
180 0.2139 180 0.1763
190 0.2336 190 0.1950
200 0.2532 200 0.2142
210 0.2730 210. 0.2345
220 0.2927 220 0.2550
230 0.3123 230 0.2740
240 0.3319 240 0.2952
250 0.3512 250 0.3162
260 0.3710 260 0.3369
270 0.3892 270 0.3575
280 0.4092 280 0.3780
290 0.4294 290 0.3982
300 0.4497 300 0.4180

Load Removed

 

Load Removed

167

 

 

 

 

SAMPLE GB SAMPLE 7B
T = -4.45°C T = -4.45°C
y. = 0.902 gm/cc Yi = 0.902 gm/cc
6 = 300 psi 0 = 300 psi
Time Deflection Time Deflection
(min.) (in.) (min.) (in.)
0 0.0150-0.022 0 0.0060
1 0.027 1 0.0104
5 0.0322 5 0.0152
10 0.0369 10 0.0192
20 0.0453 20 0.0256
30 0.0540 30 0.0334
40 0.0639 40 0.0423
50 0.0760 50 0.0526
60 0.0905 60 0.0662
70 0.1090 70 0.0829
80 0.1317 80 0.1040
90 0.1590. 90 0.1303
100 0.1912 100 0.1616
110 0.2277 110 0.1976
120 0.2674 120 - 0.2381
130 0.3095 130 0.2813
140 0.3529 140 0.3269
150 0.3992 150 0.3726
160 0.4448 160 0.4183
170 0.4932 170 0.4621
173 ‘0.5085 180 0.5049

Load Removed Load Removed

 

168

 

 

 

SAMPLE BB SAMPLE 9B
T = -4.45°C T = -4.45°C
Yi = 0.905 gm/cc Yi = 0.903 gm/cc
0 = 350 psi 0 = 400 psi
Time Deflection Time Deflection
(min.) (in.) (min.) (in.)
0 0.0078 0 0.0080
1 0.0138 1 0.0150
5 0.0219 3 0.0210
10 0.0293 6 0.0277
15 0.0368 9 0.0345
20 0.0450 12 0.0418
25 0.0549 15 0.0504
30 0.0666 18 0.0608
35 0.0815 21 0.0742
40 0.0994 24 0.0907
45 0.1218 27 0.1112
50 0.1488 30 0.1368
55 0.1801 33 0.1672
60 0.2158 36 0.2021
65 0.2542 39 0.2407
70 0.2950 42 0.2820
75 0.3371 45 0.3265
80 0.3797 48 0.3730
85 0.4210 51 0.4222
90 0.4614 52 0.4396
95 0.5010 Load Removed
96 0.5088

Load Removed

 

169

 

 

 

 

SAMPLE 10B
T = -4.45°C
Y1 = 0.904 gm/cc
First Loading 0 = 200 psi
Time Deflection Time Deflection
(min.) (in.) (min.) (in.)
0 0.0056 Change to Second Loading
0.0086 0 = 300 psi
6 0.0113 696 0.2568
10 0.0126 697 0.2587
20 0.0152 700 0.2671
40 0.0194 703 0.2768
60 ' 0.0232 705 0.2828
100 0.0299 710 0.2992
140 0.0364 715 0.3172
180 0.0428 720 0.3365
220' 0.0495 725 0.3568
260 0.0568 730 0.3772
300 0.0645 735 0.3973
340 0.0741 740 4 0.4166
380 0.0855 745 0.4360
420 0.0988 750 0.4548
460 0.1147 755 0.4736
500 0.1327 760 0.4918
540 0.1525 765 0.5101
580 0.1748 Load Removed
620 0.1994
660 0.2258

696 . ‘ 0.2508

 

170

 

 

 

 

SAMPLE 11B
T = -4.45°C
Y1 = 0.903 gm/cc
First Loading 0 = 300 psi
Time Deflection Time Deflection
(min.) (in.) (min.) (in.)
0 0.0081 860 0.3718
1 0.0128 920 0.3737
5 0.0177 980 0.3757
10 0.0218 1040 0.3775
20 0.0288 1100 0.3793
30 0.0358 1160 0.3812
40 0.0432 1220 0.3831
50 0.0519 1280 0.3850
60 0.0620 1280 0.3850
70 0.0747 1340 0.3871
80 0.0897 1400 0.3892
90 0.1085 1460 0.3910
100 0.1316 1520 0.3932
110 0.1590 1580 0.3954
120 0.1910 1640 0.3977
130 0.2272 1700 0.4000
140 0.2662 1760 0.4020
150 0.3080 1820 0.4046
158 0.3430 1880 0.4070
1940 0.4094
Change to Second Loading 2000 0.4119
118 psi 2060 0.4144
2120 0.4169
0 0.3368 2180 0.4197
5 0.3380 2240 0.4224
10 0.3388 2300 0.4250
20 0.3408 2360 0.4279
40 0.3431 2420 0.4306
60 0.3445 2480 0.4333
80 0.3437 2540 0.4362
100 0.3467 2600 0.4390
120 0.3476 2660 0.4419
200 0.3508 2720 0.4449
260 0.3528 2780 0.4479
320 0.3549 2840 0.4510
380 0.3570 2900 0.4539
440 0.3589 2960 0.4500
500 0.3608 3020 0.4595
560 0.3627 3080 0.4626
620 0.3627 2140 0.4659
680 0.3664 3200 0.4690
740 0.3683 3260 0.4720
800 0.3700 3320 0.4752

 

171

 

 

 

 

SAMPLE 11B (continued) SAMPLE 12B
T = '4.45°C
Time Deflection _
(min.) (in.) Y1 - 0.903 gm/cc
c = 250 psi
3223 8:2gig Time Deflection
3500 0.4850 (min-) (1n->
3560 0.4875 0 0.0080
3680 0.4941
5 0.0158
3740 0.4976
10 0.0194
3800 0.5009
20 0.0250
3860 0.5040
30 0.0300
3920 0.5074
40 0.0347
3980 0.5105
50 0.0395
4040 0.5139
50 0.0446
4100 0.5170
A 70 0.0500
4160 0.5203
Load Removed 80 0.0562
90 0.0628
100 0.0704
110 0.0792
120 0.0892
3130 0.1004

Load Removed

Rest 1800 seconds

0 0.0908-0.0986

1 0.1025

5 0.1086
10 0.1147
20 0.1270
30 0.1404
40 0.1556
50 0.1729
50 0.1917
70 0.2118
30 0.2333
90 0.2549
100 0.2779
110 0.3011
120 0.3254
130 0.3485
140 0.3738
150 0.4007

Load Removed

SAMPLE 13B

 

 

 

T = -4.45°C
Y1 = 0.903 gm/cc
o = 250 psi.
Time Deflection
(min.) (in.)
0 0.0086
5 0.0170
10 0.0206
20 0.0263
30. 0.0312
40 0.0364
50 0.0416
60 0.0472
70 0.0536
80 0.0606
90 0.0688
100 0.0783
110 0.0894
119 0.1006

Load Removed

Rest 3600 Seconds

0

1

5
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240

0.0910-

Load Removed

0.0983
0.1017
0.1088
0.1137
0.1226
0.1319
0.1424
0.1550
0.1699
0.1866
0.2050
0.2247
0.2451
0.2663
0.2880
0.3097
0.3314
0.3531
0.3746
0.3958
0.4167
0.4374
0.4579
0.4785
0.4988
0.5195
0.5400

 

SAMPLE 143
T = -4.45°C
Yi = 0.902 gm/cc
o = 250 psi
Time Deflection
(min.) (in.)
0 0.0068
5 0.0152
10 0.0188
20 0.0246
30 0.0298
40 0.0348
50 0.0401
60 0.0456
70 0.0520
80 0.0591
90 0.0674
100 0.0769
110 0.0878
120 0.1003

Load Removed

Rest 5400 Seconds

90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270

0.0908-

Load Removed

0.0975
0.1006
0.1110
0.1166
0.1228
0.1294
0.1372
0.1460
0.1566
0.1689
0.1832
0.1992
0.2167
0.2354
0.2548
0.2747
0.2948
0.3148
0.3348
0.3546
0.3744
0.3934
0.4126
0.4316
0.4500
0.4684
0.4864
0.5044
0.5220

 

 

 

 

T = —4 45°C
Y1 = 0.903 /cc
0 = 250 p51
Time Deflection
(min.) (in.)
0 0.0086
5 0.0171
10 0.0205
20 0.0259
30 0.0306
40 0.0356
50 0.0407
60 0.0462
70 0.0522
.80 0.0592
90 0.0671
100 0.0766
110 0.0872
120 0.0997
Load Removed
Rest 7200 seconds
0 0.0890-0.0970
1 0.1000
10 0.1088
20 0.1150
30 0.1207
‘40 0.1265
50 0.1330
60 0.1404
70 0.1494
80 0.1600
90 0.1727
100 0.1869
110 0.2028
120 0.2202
130 0.2388
140 0.2581
150 0.2780
160 0.2979
170 0.3184
180 0.3391
190 0.3598
200 0.3806
210 0.4010
220 0.4215
230 0.4414
240 0.4618
250 0.4822
260 0.5023

Load Removed

 

SAMPLE 16B
T = -4.45°C
Y1 = 0.904 gm/cc
o = 23 psi
Time Deflection
(hr.) (in.)
0 0.0013
1 0.00234
10 0.00360
20 0.00463
40 0.00620
60 0.00733
80 0.00834
100 0.00918
120 0.00980
140 0.01068
160 0.01147
180 0.01222
200 0.01319
220 0.01408
240 0.01479
260 0.01552
280 0.01612
300 0.01689
320 0.01777
340 0.01858
360 0.01940
380 0.02039
400 0.02121
420 0.02218
440 0.02288
460 0.02358
480 0.02419
500 0.02487
520 0.02509

Load Removed

 

174

 

 

 

SAMPLE 1H SAMPLE 2H
T = -4.45°C (Tensile Creep Test)
Y1 = 0.910 gm/cc T = -4.45°C
o = 209 psi Y1 = 0.909 gm/cc
02 = 74 psi
Time Deflection
(min.) (in.) Time Deflection
(min.) (in.)
0 0.009
1 0.0117 0 0
10 0.0157 1 0.025
20 0.0188 10 1.39
40 0.0238 20 1.64
60 0.0280 40 1.69
80 0.0321 ,100 1.86
100 0.0362 200 1.97
120 0.0404 340 2.04
140 0.0448 460 2.05
160 0.0496 660 2.27
180 0.0547 720 2.29
200 0.0606 1100 2.64
220 0.0670 1280 2.79
240 0.0742 1460 3.01
260 0.0821 1763 3.28
280 0.0908 2060 3.55
300 0.1070 2630 4.04
320 0.1114 2880 4.22
340 0.1234 3229 4.44
360 0.1355 3522 4.67
380 0.1489 4115 5.16
,400 0.1629 4370 5.45
420 0.1775 4872 5.87
440 0.1925 5470 6.40
446 0.2084 6273 7.38
480 0.2254 6876 7.95
500 0.2429 7602 8.60
520 0.2614 8350 9.30
540 0.2810 9146 10.30
Load Removed 10578 11.98
11240 12.80
12056 13.82
12982 14.65
14282 16.02

 

Load Removed

175

 

 

SAMPLE 3H
Biaxial Creep Test
T = -4.45°C ‘
Y1 = 0.911 gm/cc
01 = 192 p31
Time Deflection
(min.) (in.)
0 0.0074
10 0.0093
20 0.0106
40 0.0128
60 0.0147
120 0.0194
180 0.0239
240 0.0282
300 0.0330
360 0.0383
420 0.0444
480 0.0518
540 0.0608
600 0.0712
660 0.0841
720 0.0990
780 0.1162
840 0.1358
900 0.1578
960 0.1803
1020 0.2025
1120 0.2380
1180 0.2638
1240 0.2910
1300 0.3192
1360 0.3473
1420 0.3758

 

 

03 = 100 psi

Time Deflection
(min.) (in.)

0 0

5 0.02
10 0.08
15 0.10
20 0.12
30 0.18
40 0.21
50 0.23
60 0.30
80 0.39
100 0.46
140 0.56
200 0.75
402 1.50
465 1.82
580 2.59
700 3.69
730 4.02
760 4.38
830 5.28
1122 9.70
1150 10.20
1214 11.40
1246 12.02
1287 12.78
1321 13.50
1404 15.17

Load Removed

 

176

 

SAMPLE 3H: Time, Principal Strains and Octahedral Shear
Strains
Time 61 e3 82 eoct.
(min.) % %
0 0 0 0 0
20 0.0016 0.00072 -0.0023 0.0034
40 0.0027 0.0013 -0.0040 0.0058
100 0.0057 0.0028 -0.0085 0.0121
200 0.0092 0.0045 -0.0137 0.0197
300 0.0129 0.0068 -0.0197 0.0283
400 0.0178 0.0089 -0.0267 0.0385
500 0.0240 0.0102 -0.0342 0.0497
600 0.0327 0.0160 -0.0487 0.0702
700 0.0450 0.0216 -0.0666 0.0962
800 0.0609 0.0280 -0.0889 0.1285
900 0.0788 0.0362 -0.1150 0.1664
1000 0.0993 0.0443 -0.1436 0.2080
1100 0.1200 0.0537 -0.l737 0.2516
1200 0.1400 0.0623 -0.2023 0.2930
1300 0.1597 0.0728 -0.2325 0.3363
1400 0.1800 0.0828 -0.2628 0,3300

 

 

“7117117111117:(11111711111114?