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MECHANECAL PROPERTIES. OF ICE

Thesis {or the Dsgnegof M. :5; ‘
MICEEGAN STATE UNBIERSETY
Thomas R, Halbrook.

1962

3142515

This is to certifg that the

thesis entitled

Mechanical Properties Of Ice

presented by

Thomas R. Halbrook

has been accepted towards fulfillment
of the requireménts for

_LS;__ degree in __Q:_E_.__

(3‘ E: Owflww

Major professor

 

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University

 

 

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IECHANIGLL PROPERTIES OF ICE

By

Thomas R. Halbrook

AN'ABSTRACT

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

MASTER OF SCIENCE

Department of Civil Engineering

1962

ABSTRACT
MECHANICAL PROPERTIES OF ICE
by Thomas R. Halbrook

This thesis is an investigation of the mechanical
properties of polycrystalline ice in compression and in
creep.

The investigation consists of the determination of
the compressive strength, modulus of elasticity, an analysis
of the minimum.and tertiary creep rates, and a study of the
deformation characters of the material.

Confined and unconfined triaxial tests were used to
determine both stress-strain relationships and creep curves
for identical polycrystalline ice specimens at ~40 C. The
effect of rate of axial deformation was varied in constant-
rate-of-strain experhments and creep under constant load
were conducted under several large loads.

It was found that by increasing the rate of axial
deformation, polycrystalline ice will exhibit larger values
for'Young's modulus and a larger ultimate strength.

V The creep of ice under large loads is greatly
influenced by stress level and confinement. Bulk flow laws
and simple viscoelastic models will give only rough approxi-

mations of behavior. Confining pressure appears to increase

Thomas R. Halbrook

the minimum.creep rate and to decrease the tertiary creep
rate. Based on strain.measurements and the observed in-
fluence of confining pressure, it is likely that tertiary

creep occurs after the ultimate strain of the material has

been exceeded.

MECHANICAL PROPERTIES OF ICE

By

Thomas R. Halbrook

A THESIS

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

MASTER OF SCIENCE

Department of Civil Engineering

1962

ACKNOWLEDGEMENTS

The author wishes to express his deep gratitude for
the guidance and inspiration of Dr. 0. B. Andersland,
.Assistant Professor of Civil Engineering, Michigan State
University, under whose supervision this study was con-

ducted.

Special thanks are due Mr. John‘w. Hoffman, Director
of Engineering Research, Michigan State University, for the
equipment made available through his office; and to Dr.
George Mass, Professor of Applied Mechanics, Michigan State
University, for his valuable advice concerning the visco-
elastic properties of engineering materials.

The author is indebted to his colleagues, B. N.
Beyleryan, A. K. Loh, and R. W. Christensen, for their
thoughtful suggestions and help in the testing and analysis.
Through the efforts of Mr. Leo Szafranski, our departmental
mechanic, costly time delays were avoided; the author is
grateful for his help.

Finally, the writer would like to thank his wife for
typing the draft and final copy of the thesis. Without her
warm encouragement and understanding this thesis would not

have been possible.

11

TABLE OF CONTENTS

MUWEMENTSOOO'OeOOOeee

LIST OF
LIST OF
Chapter
I.
II.

III.

mm 0 O O O O O O O O O O O

FIGURES . .

INTRODUCTION . . . . . . . . .

MECHANICAL PROPERTIES OF ICE, LITERATURE

mm C O O O O O O O O O O

Compressive Strength
Tensile Strength . .
Shearing Strength .
Elastic Properties
Creep

Primary Creep .

Secondary 019001) e e e e e
Tertiary creep e.e e e e e
Activation Energy . . . .
Effect of Confining Pressur

Viscoelastic Properties . .

APPARATUS AND PROCEDURES . . . . .

EXPERIMENTAL RESULTS AND DISCUSSION

Results of Constant Rate of Axial Defor-
nation Experiments 0 e o e e e e e e
Results of Creep Experiments . . . .

Minimum Creep Rate Observations

Tertiary Creep Observations

CONCLUSIONS . . . . . . . . .

iii

Page
11

vi

65
80

84
90

99

RECOMMENDATIONS FOR FUTURE WORK . . . . . . . . . . . 101
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 102
APPENDICES . . . ... . . . . . . . . . . . . . . . . . 105
A Detailed Procedures . . . . . . . . . . . . . . 106
B Data . . . . . . . . . . . . . . . . . . . . . 108

iv

LIST OF TABLES

Table Page

1. Summary of Previous Investigations of
SooondaryCroop-....o..........31

2e Activation.Energy of Ice . . . . . . . . . . . . 56
3. Summary of Constant-Rate-of-Strain Results . . . 73
4. Summary of Creep Results . . . . . . . . . . . . 86
5. Seating Strain Observations . . . . . . . . . . . 109

LIST OF FIGURES

Figure Page
1. Structure of a Frozen Soil . . . . . . . . . . . 3
2. Stress on an Ice Crystal in.Frozen Soil . . . . .
3. Molecular Structure of Ice . . . . . . . . . . . 7

4. Unconfined.Compressive Strength of Ice Versus
Tmperfituro.......o..........12

5. Average Tensile Strength.As a Function of
Load Application . . . . . . . . . . . . . . . 16

6emioncroepcurVOeeeeeeeeeeeeeee23

7. Variation of Minimum Creep Rate With
Compressive Stress .. . . . . . . . . . . . . . 28

8. Variation of Creep'fiith.Shear Stress . . . . . . 34

9. Variation of the Activation Energy of Ice
With Stress O O O O O O O O O O O O O O O O O O 39

lO.~ Influence of Confining Pressure on Creep . . . . 41
11. Basic Viscoelastic Models . . . . . . . . . . . . 43

12. Behavior of a Burgers Model in Creep and
Rel-“3t 10D 0 O O O O 0 O O O O O O O O O O O O 46

13. Disassembled.Aluminum Ice Mold, Specimens, and
Loading P‘rapharnali‘ e e e e e e e e e e e e e 52

14. Schematic Diagram.of Ice Specimen Prepared for
TOGtingeeeeeeeeeeeeeeeeeeee55

15 0 Tea t Equimant O O O O O O O O I O O O O O O O O 58

16. Schematic Diagram of Hydraulic and Cooling
SYBtmeeeeeeeeeeeeeeeeeeeeeo

17. Temperature Measuring Equipment . . . . . . . . . 63

vi

Figure
18.
19.

20.
21.

22.

23.
24.

25.

26.

27.

28.

Page
Typical Appearance of Specimens.After Testing 66

Typical Response of Ice Subjected to a Constant
Rate of Axial Deformation . . . . . . . . . . . 68

a, b, c, d, Stress-Strain Curves for Ice . . . . 69

Variation of'Young's Modulus of Ice With Strain
Rateeeeeeeeeeeeeeeeeeeeee75

Variation of Ultimate Strength of Ice With
Str‘m Rate 0 O O O 0 O O O O O O O I O O O O O .77

Mohrs Circle for Polycrystalline Ice . . . . . . 79

a, b, c, Creep Curves of Ice Under Constant
Lea

O O O O O O O O O C O O O O O O O O O O O 81

Variation of Minimum Creep With Load, Time and
Confinment..................8'7

Variation of a Function of Tertiary Creep Strain
With Time and Confining Pressure . . . . . . . 91

Variation of the Exponent of Time for Tertiary
Creep With Load and Confinement . . . . . . . . 93

Comparison of a Typical Stress-Strain Curve to
ITypIOSICI‘OOpCurYe.............95

vii

CHAPTER I
INTRODUCTION

During the last ten years a great deal of attention
has been given to the mechanical preperties of ice. Since
the early 1950's, the United States Army Corps of Engineers
have added muoh.knowledge to the field of ice mechanics
through their studies pertaining to tunnels and cavities
built in the Greenland Ice Cap. Many other organizations
are interested in the behavior of ice as it relates to the
preperties of frozen soils. Prior to this time much of the
investigation was conducted by geologists searching for
answers to the riddle of glacial movement.

The work done to date concerns itself primarily with
the behavior of ice under low stress. Knowledge of ice
mechanics under high stress is practically nonexistant, and
very little is known about the effect of confining pressure.
It is reasonable to expect that ice contained in soil pores
will be subjected to high stresses, values approximating
pressures between point contacts of individual soil particles.

All the water in a frozen soil mass is not frozen.
Substantial proportions do not freeze at temperatures as low
as -20° C. (Lovell, 1957). The amount of freezing that occurs
depends upon the temperature, soil type and arrangement of

the soil particles (Jackson and Chalmers, 1957: Chalmers, 1960)

1

and the ions in the absorbed.water layer surrounding the
soil particles (Leonards and Andersland, 1960).

Figure la shows a typical structure of a claylike
soil. The channels and interstices (voids) of the mass may
be filled with.water and/or air, while the amount of water
is referred to as the moisture content. Nucleation can only
occur when the radius of curvature of the water within the
interstices and channels has a magnitude greater than the
critical radius for the prevailing temperature (Jackson and
Chalmers, 1958). Ice will form in the larger interstices
before it forms in the channels as shown in Figure lb. At
a given temperature the ice structure within a soil mass is
composed of many individual crystals of ice that are poly-
crystalline in nature (Chalmers, 1960). It is reasonable
that the prOperties of polycrystalline ice contained in the
soil pores play a major role in the shear strength of frozen
soil.

It can be shown that the ice within the interstices
is subjected to large pressures using the analysis given by
Skempton (1961). With reference to Figure 2, consider a
particle of ice in contact with a particle of soil on some

statistical plane, A and occupying a gross area, A, in a

.9
plane parallel to this contact. Skempton (1961) defines the
contact area ratio, a, as follows:

a =1§E 3 .ZaL<: 1

If the force normal to the contact plane is P and the shear

3

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force is T, the total normal stress, 67, and.the shear stress,

'7, are
P - T
0.31- and T‘I.
Stresses at the interracial contact are
P T
67 = and =
s *1: 7; 3: °

If a 9.: 0.1, then a; would be ten times 5', a fairly large
stress. The unfrozen pore water contributes an additional
amount of pressure, u. In kaolinite clays this unfrozen water
at a distance of 10 X from the soil particle can withstand a
normal stress of 25 atmospheres (Martin, 1959).

For equilibrium.normal to the plane

P=Pa+(A-A.) ue

Hence,

 

=e - a. u
0' aJ8+u a)u and era-Jar.“ .

Since a is less than one, it follows that the contact stress
is somewhat larger than the gross stress acting on the
section.

The purpose of this thesis is to provide a platform
for more extensive work in the field of frozen soil.mechanics,
primarily through.a review of existing literature on ice and
several series of experiments on polycrystalline ice. The

influence of varying constant rates of deformation has been
investigated. This has often'been used to explain the vari-
ability of experimental results. Temperature is made a
constant (-4° C). The creep properties of ice have been
investigated in the past, however only in relatively low
stress ranges. A limited investigation of high stress creep
has been conducted. The above two series are subjected to a
third variable, confining pressure, in order that its effect

may be better understood.

CHAPTER II
IECHANICAL PROPERTIES OF ICE--LITERATURE REVIEW

The mechanical properties of ice are related to the
molecular structure of an ice crystal. Figure 3a shows the
atomic arrangement of a single 320 molecule (Bjerrum, 1952).
The atoms arrange themselves in ”puckered hexagonal layers"
(02. cit.) as shown in Figure 3b. The oxygen atoms are
‘ shown as black dots and the bonding provided by the hydrogen
atoms are the black connecting lines. The number of bonds
perpendicular to these layers is small (Figure 3b); and it
is along these planes that slippage occurs. These planes
are called gliding planes of basal planes.

Glen and Perutz (1954) and, independently, Steinemann
(1954) reach.thc conclusion that only along these basal planes
does gliding take place during creep. Their conclusion is
supported by testing ice crystals with varying tilts of the
basal planes and by x-ray defraction studies of the crystals.

The method of preparing or obtaining ice crystals can
be done in several ways. Single crystals of reasonably good
quality can be grown using a technique perfected by Landauer
(1958). Large single ice crystals have been taken from
glaciers, but obtaining and transporting them is difficult

and expensive.

 

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Preparing single crystals of ice for testing usually
requires forming a specimen into a given shape, (cube, cyl-
inder, hollow cylinder, etc.). Stresses introduced on the
surface of the specimen.during preparation are thought to be
a prime source of the erratic behavior of the results in ice
experiments (Jellinek, 1957; Nakaya, 1958; Butkovich, 1954).
SIPRE1 invegtigators have in the past prepared.samp1es by
sawing blocks and beams from larger ice samples, trimming
them on a lathe, sandpapering and finishing them to a high
gloss with.silk, and planing them with a carpenter's plane.

Polycrystalline ice samples, as compared to single
crystal samples, are more readily prepared. Duplicating the
samples is important, with single crystals this is extremely
difficult to accomplish. The prOperties of an aggregate
polycrystalline sample are not always simply statistical
averages of the properties of a single crystal taken over
all orientations. While this is approximately true of
properties which depend primarily on the bulk structure, like
the elastic modulus, it is not necessarily true of plastic
phenomena (Hill, 1956).

Many investigators have pointed out that air in the
water causes voids in the ice samples, altering the physical
properties and characteristics. This is overcome by ''de-

airing” the water by boiling it beforehand.

 

1SIPRE refers to the United States Army Corps of
Engineers' Snow, Ice and Permafrost Research.Establishment

at Wilmette, Illinois.

Leonards and.Andersland (1960) prepared ice cylin-
ders in an aluminum.mold by supercooling distilled, de-
aired water. These samples required trimming only on the
cylinder top. Their procedure allows many samples to be
prepared under identical conditions.

When ice is frozen by supercooling (specially puri-
fied water can be cooled to below -200 F before freezing
occurs) two distinct processes occur-~nucleation and growth.
Nucleation is the initiation of freezing that takes place.
below 32° F, and the subsequent freezing that occurs is
referred to as growth.(Chalmers, 1960). When nucleation
occurs, latent heat is given off into the surrounding water,
which is then warmer. Further cooling again brings the
water below the freezing point and the growth.process begins.
Ice crystals formed in layers have a dendritic shape and
continue to grow on their basal planes as heat is removed
through conduction.

Steinemann (1954) discusses physical characteristics
of ice formed by supercooling at -1° 0 and -5° c. In the
first case many faults in the linear structure form, but
few sub-microscopic faults exist. Ice formed at -5° 0
always led to the growth of nearly perfect crystals in every
direction.

Compressive Strength
The compressive strength of ice has been extensively

investigated. An excellent summary of the results prior to

10

1951 may be found in the companion volume of SIPRE Report
Number 8 (Volume II, Appendix B, Table 12). The samples
ranged in source from poor quality pond ice and various
layers of river ice to artificially prepared ice. The major-
ity of the tests were performed on cubes or blocks. Many of
the reported investigations did not state the shape of the
test sample. .

According to the SIPRE table, Vitman and Shandrikov
(1938) were the only investigators who studied the strength
by means of constant rate of strain (0.1 mm per second).1
The tests were performed on river ice cubes 1.97 inches on a
side. Many preferred to investigate compressive strength
from a constant rate of stress increase (psi/minute), however,
this information is lacking in numerous cases.

Vitman and Shandrikov's results show a general in-
crease in the strength of ice as the temperature is decreased.
However, they report a rather sizeable variation in the
strength observed. At -4° c (24.e° F) an interpolation of
their results shows that the average strength would be about
180 psi with a maximum of 290 psi and a minimum of 100 psi.
Brown (1926) reported a similar increase in strength corre-
sponding to lower temperatures, but his results as to magni-
tude of strength are not in agreement with those found by
Vitman and Shandrikov. This strength variation is believed
to take origin in the methods used to prepare and test the

 

1Corresponds to a rate of 12% of the original sample
height per minute.

11

specimens.

In summary, no definite correlation between tempera-
ture and compressive strength, in fact, not even a definite
average compressive strength can be established fran the
results of investigations prior to 1951. The Frost Effects
Laboratory, which.performed.the‘work in SIPRE Report Number
8, found that the following variables have a profound in-
fluence upon the compressive strength of ice:

1) A sample having a number of small crystals has a
greater strength than.a sample having one or two
crystals.

2) The unifonmity of loading has an effect upon the
failure mode whicb.may occur as one unit or as a
group of columns. Columnar failure was not
uncommon.

3) At constant temperature, a very rough trend
toward increased strength with increased rate
of loading is indicated.

T. R. Butkovich (1954) performed a series of tests on
cylindrical ice specimens to determine the ultimate strength
of ice. He used lake ice, natural snow ice, and commercial
ice. The specimens had a widthrto-height ratio of l to 3,
which he states is a recommended ratio. His review of lit-
erature shows that after extensive testing, during the build-
ing of the Southern Manchurian Railway, the Japanese had
formulated empirical relationships for crushing strenth.which

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13

showed a linear increase of strength with decreasing tempera-
ture. However, their results were highly scattered.

His results from ice in compression showed conclusive-
ly that temperature had a pronounced effect on ice. A typical
curve is that of clear lake ice from Portage,Michigan, shown
in Figure 4. An important conclusion is that the strength is
indeed dependent upon how the load is applied to the ice
sheet itself. Again, a wide variation in strength was ob-
served. His results showed a variation of as much as
140 psi from the mean strength. The mean was obtained from
the results of ten samples. All samples were tested in
unconfined compression, loading with a rate of approximately
5 kg/omz-sec. The specimens were loaded.manua11y and the
failure progression was described as follows:

When the load was applied parallel to the length of the
prismatic crystals small cracks began to appear inside
the ice specimens. They increased in size and number,
particularly Just before breaking. The time when the
cracks first appeared was different for loading normal

to ths candles, and the cracks first appeared at an angle
of 45 near the ends. Chamois skin or sponge rubber was
used to seat the ice samples in the crushing apparatus.
Temperatures were accurate to:1.0o C.

In 1960 Leonards and Andersland found that the
strength of ice increases approximately 15 psi per degree
centigrade in their unconfined compression tests using
identical, polycrystalline cylindrical samples. Their results
showed good agreement with those of Butkovich (see Figure 4).

Interpolating from their curves, ice formed in this

manner at -4° 0 Mean average strength of about 570 psi with

14

a.maximum of 430 psi and a minimum.of 510 psi. They used a
constant rate of strain throughout the series of 2% of the
original height per minute.

Tensile Strength

Previous to 1954, very few measurements of the tensile
strength.of ice were made. The information was computed from
observations of other prOperties, such as the bending of ice
beams. Butkovich (1954) pulled cylindrical ice specimens in
an electro-hydraulic tensile testing machine. He found a
slight increase of tensile strength with.decreasing tempera-
ture for large grain commercial ice. Because his tests were
of sizeable specimens, his results are called bulk tensile
strength.

Butkovich's technique for holding the specimen during
tension testing was to grip the specimen with.a wire. Jellinek
and Brill (1956) improved the procedure by freezing the spec-
imens to a roughened.meta1 plate.

In later, more complete, studies of tension by
Jellinek (1957), great care was taken to adhere the ice to
polished stainless steel. He found definite indications
that there were certain stress concentrations at the ice-
metal interface, eventually causing failure at this Joint.

0n large samples he found that the tensile strength
was affected by rate of loading (Figure 5). His explanation
of this behavior is:

At low speeds of loading, an appreciable plastic flow
takes place. Hence, an appreciable lateral stress component

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16

will be present in the neighborhood of the interface,
and though counteracted by a certain amount of stress
relaxation, (Ed. this stress component) will lead to
smaller tensile strength values. At higher speeds of
loading, less plastic flow will take place and the lat-
eral stress component will decrease. Therefore, the
tensile strength should be higher than at relatively low
rates of loading. However, at the high rates the stress
relaxation will be small, which in turn will tend to de-
crease the tensile strength. Hence, there will be an
optimum.region of loading rates Where all these effects
will combine to give a.maximum tensile strength . . .
However, When Jellinek tested very thin ice discs,
different characteristics were observed. A complete cohesional
break occurred (the larger samples seemed to partially “peel"
off their mountings) and the tensile strength was found to be
a function of volume, thickness, and diameter of the disc.
However, this apparent correlation between strength and the
height-to-diameter relationship, based on the assumption that
the tensile strength changes with geometric considerations,
was found to be untenable. By a statistical analysis
Jellinek concluded that this behavior of the discs is best
explained by the imperfections in the ice. He assumed that
there is a definite distribution of imperfections, and that
these imperfections can stand stresses up to a critical stress,
which, if exceeded, causes a crack to open and rupture occurs.
In the larger samples the whole distribution of imperfections
will be represented from the weakest to the strongest; in the
small specimens only those in the neighborhood of the distri-
bution maximum will predominate.
His analysis showed good agreement with the experi-

mental model, and he concludes that tensile strength is

17

largely governed by the imperfections in the small specimens.
In the case of large specimens, the failure near the inter-
face was due to stress concentrations and imperfections. He
states that since his statistical method is only an approxi-
mation, there is the possibility that a stress distribution
effect is present.

§hearing Strength

The nature of ice in creep alone seems to indicate
that shearing forces play a.major role in the behavior of
ice. To date, however, only one attempt to measure the
shearing strength of ice can be located, and the reliability
of it is questionable.

The Frost Effects Laboratory (1952) attempted to
determine the shearing strength by use of Mohr diagrams. To
do so they used tension results and compression results
without confining pressure, and their results must be thought
of as temporary resistance to shear (units are psi/min).
Criticism.of their work (Plate 76), in view of more recent
findings, would include:

1) Rate of stress application for tension specimens
was 1/10 of the rate of that for compression
specimens. Consistently through the years the
effect of varying rates of stress application
has been pointed out as having some influence
upon the strength of ice.

2) The specimens themselves varied both in

18

crystalline structure and method of forming the
samples. Stresses produced in the samples
during trimming could have been large.

3) Tension failures gave poor results; failures near
the gripping surface occurred.

4) Results used to produce the average values varied
from maximum.to minimum as much as 680 psi--a
value greater than the average strength itself.

5) Two values of ¢>, the angle of internal friction,
are obtained, apparently dependent upon the tem-
perature at which the test was conducted. Their
calculations of shear strength use ¢ = 30°.

It is interesting to note that comparable results
were obtained between the shear strength based on a Mohr's
envelOpe with (b = 30°, and the strength obtained from direct
shear tests.

Elastic Properties

Because ice is not an elastic material, but rather a
viscoelastic material, the concept of ”elastic" properties
of ice are misleading. However, some of the previously
determined values are noteworthy.

In 1928, Voigt described the elastic behavior of
single ice crystals as being a summation of six elastic
moduli and/or six elastic coefficients by the following
relationships:

0' == c 51 and E =£E: s 5r
iélikk inglikk

19

671s the stress, 6' the strain, Cik the elastic moduli and
31k the elastic coefficients. Stephens (1958) reports the
results of several investigators who by various sonic methods
have determined these elastic constants. As ice is hexagonal,
there are five independent elastic constants (01k or 81k).
This may be verified.by the relationships between the elastic
moduli and coefficients.
6

Z 1f0r1=J
°ik°ik=

n=1 0 for i = J

Because ice does have viscoelastic properties, dynamic
methods give the only reliable results.

Stephenie summary of the work done shows that temper-
ature does affect the elastic constants, but only in a very
small degree. This is to be expected, the values of elastic
moduli of crystalline solids will tend to decrease slightly
as their melting point is approached (Jastrzebski, 1959).
Stephens reports the values of the elastic constants for ice.

The value of YOung's modulus determined from.these
values will vary with the crystal arrangement (strains or
stresses in the kth'direction must be observed or computed
respectively). Variations in experimentally determined
values of E are explained by considering that E depends upon
the preferred orientation of the grains (Van Vlack, 1959).
The‘Young's modulus of specimens prepared with random orien-

tation will represent an approximate average value.

20

The review of the literature completed by the Frost
Effects Laboratory for SIPRE (1952) shows that investigations
of YOung's modulus for ice have given comparable results
since 1885. As the temperature decreases, the value of
‘Young's modulus increases slowly. Their linear relationship
gives E of 1.25 x 106 psi at 52° F and 1.52 x 106 psi at -54°F.
The values were obtained from both sonic and static test meth-
ods. The static methods were primarily beam flexure tests.

It is important to realize that these values represent
the initial tangential modulus, and'will be higher than the
secant modulus which is conventionally used in engineering
design problems.

Nakaya (1959) observed a hysteresis phenomenon in the
relationship between Young's modulus and temperature. A
similar effect is produced in.metals when they are hardened
by quenching. The values of E measured as a specimen cooled
were found to be lower than the values measured from the
same specimen as it was allowed to warm. As the temperature
of the warning specimen approached the temperature at which
the experiment began, the values of E warming approached the
earlier values of E cooling producing the hysteresis curve.

The density of ice has a pronounced effect upon 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 showed for

densities between 0.915 and 0.903 the Young's modulus varied

21

from 1.55 x 106 psi to 1.09 x 106 psi respectively. He used
a sonic technique and his results agree well with those
previously found by other investigators.1

The rate of stress application has a pronounced effect
upon the elastic properties of ice and is thought to be caused
by the viscous nature of ice. The influence of this viscous
nature was also noticed in tests on snow ice beams used to
determine the elastic modulus (Halvorsen, 1959). Halvorsen
found that his values of E varied between 175,500 psi and
194,000 psi (1.22 x 104 kg/cm2 to 1.35 x 104 kg/cmz) as the
rate of load application varied. The higher values were
obtained at faster loading rates.

Poisson's ratio, ,u , has often been calculated for
ice with.values ranging from 0.29 to 0.565. Recent investi-
gators (Jellinek, 1959; SIPRE Report 8, 1952; Jellinek and
Brill, 1956) have preferred 0.50 as the representative
Poisson's ratio for ice. The value of Young's modulus, in
conjunction with either Poisson's ratio,}/ , or the bulk
modulus, B, or the shear modulus, 0, may be regarded as
fundamental and sufficient to calculate the other two pro-

perties (Jastrebski, 1959).

 

1Using resonant frequency to determine Young's modulus
has one drawback; the modulus for many viscoelastic materials
has been found to be quite sensitive to the frequency at which
it is measured. Nakaya (1959) reports that although this fre-
quency dependence is complicated, the effect is not very great
for ice. His values for E were corrected for frequency
dependence.

22

Creep

The relation between stress, strain, and time for
ice has been investigated by many authors. Creep has been
observed at very low stress levels (Jellinek and Brill, 1956).
Because creep is influenced by temperature, the problem of
relating creep behavior by bulk flow laws is complex.

Single ice crystals behave somewhat like polycrystal-
line ice (Butkovich and Landauer, 1959; Jellinek and Brill,
1956). Butovich and Landauer (1959) find that the scatter
of results between different types of ice specimens is comp-
arable to the scatter of results between specimens of the
same type.

A typical creep curve is shown in Figure 6, in which
the three basic stages of creep are illustrated (Finnie and
Heller, 1959). Primary creep of ice consists of an initial
elastic strain (which can be recovered after unloading) and
an initial inelastic strain (which is not recoverable).
Following these initial strains is a transient stage in
which the creep rate rapidly decreases, eventually slowing
down to a minimum, which is called the secondary creep stage.
Tertiary creep, the third stage, occurs at a faster rate
than the secondary or minimum creep rate. This behavior is
thought to be a result of intercrystalline movement as
Opposed to the intracrystalline movement of primary and
secondary creep rates. Tertiary creep can occur at any

temperature providing the stress level is high. In metals,

23

 

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24

near their melting points, tertiary creep has occurred under
low stresses (Finnie and Heller, 1959).

The bonds between the puckered layers of ice are
logical locations of gliding. eray diffraction patterns
of single ice crystals show that gliding appears to take
place along the basal plane (Glen and Perutz, 1954; Steine-
man, 1954). Recently, Kemb (1961) has shown through a
thoughtful investigation of previous material and a detailed
argument that this is the only logical direction of glide.

Tertiary creep of single ice crystals occurs only
after a strain of 10 to 20 per cent (Steinomann, 1954) (See
Figure 8). X-ray diffraction patterns show that asterism
(smearing of spots) does not increase appreciably until the
crystal is subjected to strains over 10 per cent (Glen and
Perutz, 1954). Kamb (1961) concludes that the bonds between
the layers become broken after large strains and gliding
takes place at a higher rate.

Much discussion has been devoted to the creep curve
of ice. Although there exists a large amount of data on the
subject, it is to some extent contradictory. The observed
behavior of ice in creep is best presented by considering
the different stages of creep. Results on single ice crystals
and polycrystalline samples are presented so that comparisons
can be made between them. I
Primary Crepp

Jellinek and Brill (1956) investigated creep of single

25

ice crystals and polycrystalline ice in tension. Aware that
high stresses would produce necking in their specimens (and
thereby stress concentrations), they kept their range of
stresses between 5 psi and 55 psi. By removing the load from
a sample in the second stage (minimum creep rate), they found
that the initial elastic recovery was of the same magnitude
as the initial elastic deformation. Since total recovery was
not possible, they concluded that during load application not
only elastic deformation but plastic flow occurs.

If this plastic flow was considered to be linear with
time, they found it could be represented by coefficients of
viscosity that were apparently independent of stress. These
values also agreed with the viscosity coefficients found from
the minimum.creep rates. They further concluded, then, that
Newtonian flow had occurred.

Secondaryggreep Stagp, The Minimum.Rate of Creep

Bulk flow laws have usually been derived for the
minimum.creep rate of ice. Three basic equations have been
used to represent the bulk flow of ice (Butkovich and Lan-
dauer, 1959). They are

€=ctr................(1)

X=Asinh(7;/'r) ..........(2)

é=kn ...............(5a)~

or

i=1? ................(5b)
Jellinek and Brill (1956) and Griggs and Coles (1954) use

26

a fourth form combining the features of equations 1 and 5a as.
6 = ad-ntm O O O O O 0 O O O O O O O O O (4)

The first equation predicts a linearity of log strain
versus log time; r represents the slope of the line in such
a plot. Butkovich and Landauer (1959) point out that a creep
law of this form is at best only a rough approximation, as
stress level plays a.major role in the flow of ice. Equation
2 is not recommended by Butkovich and Landauer (1959) because
it was awkward to handle and did not give a good representa-
tion of data. Equations 5a and 5b assume a linear relation-
ship between time and strain. In the past, most investi-
gators have preferred these forms to represent the creep of
ice under low stress. Butkovich and Landauer (1959) inves-
tigated all four equations and feund that equation 5b best
suited the results of 160 specimens.

Glen (1955) carried out compression experiments on
polycrystalline ice in a stress range from 10 psi to 154.8
psi, studying the steady-state creep rates over long periods
of time (100 hours or:more). He found that (see Figure 7)
by plotting the logarithm of the minimum compressive strain
rate, 6 , against the logarithm of stress, 0', a nearly
straight line was produced for each temperature inspected.

As long as the temperature was near the melting point the

value of n (Eq. 5a) had a value of 5.17. At lower tempera-

tures n became 4.2

27

Steinemann (1954) performed tests in a Baush.Appa-
ratus which.allowed direct shear to be placed on his single
ice crystals. Since ice glides along its basal plane, he
felt that shear and shear rate (equation 5b) would produce
the most reliable results. His analysis was like that of
Glen (1955) and he found n to have a mean value of 5.2,
comparable to Glen's results (see Figure 7).

The constant k, according to Glen (1955, 1958) varies
with the temperature. It will not affect the rates of strain-
ing (6" and 8" ), but it will affect the magnitude of the
strain. Steinomann (1954) believes values of k will be
dependent on stress.

Butkovich and Landauer (1959) using numerous flow
data on polycrystalline ice, found n = 2.96 for Eq. 5b.
Although their strain rate became apparent only under a log
strain log time plot (a contradiction to Eq. 5b), their
results were best represented using this form. 0n single
ice crystals they found n = 2.5.

The cause for variation between flow laws for poly-
crystalline ice as compared to single ice crystals is not
known exactly. Steinomann (1954) expects that the value of
n will be comparable, and that variations in the magnitude
of k will account for the differences. Butkovich and Lan-
dauer (1959), on the other hand, believe that the variation
in polycrystalline ice is because deformations are not

occurring on the basal gliding planes of individual ice

   
 

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29

crystals, but rather at the grain boundaries.

Higashi (1959) found that the minimum creep rate was
not proportional to a constant power of stress. From his
results of experiments measuring the plastic deformation of
hollow ice cylinders subjected to hydrostatic pressure, he
concludes that n in Eq. 5a and 5b is an increasing function
of stress. In scrutinizing Glen's (1955) results, Higashi
(1959) found the same behavior (see Figure 7). Jellinek and
Brill (1956) found serious discrepancies between their work
and that of Glen (1955). They felt that Glen's results
should correspond to plastic flow, the differences, they
explain, are probably due to the methods of testing (tension
versus compression; smaller stresses versus higher stresses).

Jellinek and Brill (1956) base their viscoelastic
analysis on results which fit Eq. 4. Several relationships

become apparent upon inspection of Equ 4, as follows:

A series of creep experiments conducted at different
stress levels allows the isolation of time by plot-
ting strain values versus stress at constant times.

This is called the techniques of variable isolation.

The exponent of the time variable, m, can be obtained

by allowing a function of stress and strain to become

a parameter in itself. This is accomplished by divid-

ing the strain by the stress raised to the nth power

50

and plotting the logarithms of 5 / J n versus the

logarithm of time. m will be the slope of such a

line.

Jellinek and Brill (1956) found that their stress-strain
curve drawn in this fashion produced a weries of straight
lines, i.e., the exponent n was equal to one. Another
series showed this plot to be curvilinear, and when replot-
ted on log-log paper a value of n = 1.4 was obtained.1

They found m.to be approximately 0.47 for both
series of experiments.

Glen (1955) also used a form similar to Eq. 4. He
analyzed his creep curves by assuming that they fit Andrade's
Law. Although the results were scattered, he found that the
exponent of time would have a value of approximately 1/5
(see note in Table l).

Griggs and Coles (1954) performed compression tests
on single ice crystals and found progressively accelerating

creep. Their empirical relationship also includes the dif-

ference in temperature from 00 C, as follows:
E=B[(J-bT02)t‘Jzeeeeeee(5)

For a constant temperature this reduces to a form similar

to Eq. 5a, or

 

1A linear preportionality must exist between stress,
strain, and time for a viscoelastic analysis. Jellinek and
Brill used the portion of their second curve that was nearly
linear.

31

 

 

 

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32

6 = adfizt2

*’ 2
02‘- 0“th .

Their work was performed over a relatively short period of
time, which Glen (1958) feels is responsible for the lack of
a uniform flow rate.

A summary of the results of the different investigators
is in Table 1. The variables which influence their results
include:

1) Tension creep versus compression creep

2) Shear stress versus compressive stress

5) Polycrystalline ice versus single ice crystals

4) Stress level

5) Duration of loading.

In light of Jellinek's (1957) findings, where phy-
sical behavior apparently is highly dependent upon the imper-
fections in the crystal lattice, it follows that local micro-
scopic deformations would not have a pronounced effect on
polycrystalline ice. Properly mounted, compression and
tensile polycrystalline samples shopld produce comparable
results.

Tertiary Cregp

Steinemann (1954) observed tertiary creep in single
ice crystals and referred to it as "softening" of the crystal.
He is the only investigator to date to calculate a bulk flow

for tertiary creep (n = 1.5), although most investigators have

35

observed it. His results are shown in Figure 8. Glen (1958)
points out that in compression, tertiary behavior was observed
at stresses higher than 4 kg/cmz (57 psi).

Previous investigators (Glen and Perutz, 1954: Jel-
linek and Brill, 1956; Glen, 1955, 1958; Butkovich and Lan-
dauer, 1959; Kamb, 1961) have hinted that tertiary creep
occurs after the bonds between the individual crystals have
broken. In polycrystalline metals the grain (or crystal)
boundaries represent a concentration of dislocations, imper-
fections in the crystal lattice structure (Finnie and Heller,
1959). When the polycrystals are subjected to large strains,
stress concentrations occur at the many grain boundaries and
voids begin to open. Creep results (where slower rates fa-
cilitate observation of intergranular failure) have shown
that the interstitials (extra atoms in the atomic lattice
work) migrate to these voids. Eventually a point is reached
where slippage between the crystals is excessive, macro-
sc0pically defining the ultimate strength of the material.
Finnie and Heller (1959) suggest that tertiary creep may
begin at this point.

Activation Energy of Ice

The atomic structure of materials may be disturbed
and distorted by'a very small amount of energy, in fact,
much less than that of the bonds holding the molecules to-
gether. To bring about a disturbance so that a distortion

will occur requires an.amount of energy called the activation

 

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35

energy, A H (Finnie and Heller, 1959). Rates of chemical
reaction and physical behavior of engineering materials are
governed by the activation energy criterion, which is highly
dependent upon the temperature of the material.

Many investigators have shown that the creep rate of

a material is governed by activation energy, as follows

éerAH/RTeoeeeOOO(6)

where R is the gas constant, T is the absolute temperature,

A is a constant, and e is the base of natural logarithms.
Investigators of nickle and aluminum have indicated that the
activation energy for these metals is dependent upon the stress

level, leading to

é=AeAH+f(a)/RT e e o e e ('7)

where "f(CT)” means function of stress. Activation energy
is usually expressed as calories per gram mole of a system.
On a semilog plot, the log of creep strain rate
versus the reciprocal of the absolute temperature will show
a nearly linear relationship. The energy of activation for
this viscous flow,A H, can be determined from the slope of
the curve within a particular temperature range (Jastrzebski,
1959).
As may be seen from Table 2, investigators find an
activation energy between l5k cal[mole and 57k cal[mole for

k cal

ice. Glen (1955) reports a value of 51.8 /mole.

36

Landauer (1951) points out that this value [is strongly

dependent upon the use of one creep specMen at -l2.8° C,

and a liberal error could be present in the value. Landauer

TABLE 2
ACTIVATION ENERGY OF ICE

 

 

 

 

 

 

Activation
Investigator Specimen (Test Type Temp. Range Energy
k cal/mole
Glen, J.W. poly. compres- 1.5 to
(1955 ) ice s ion 12 .8 51 .8
Jellinek and poly.
Brill (1956) ice tension 5 to 15 16.1
Landauer.» compres- 5.6 to
(1957) snow sion 15.55 14 to 27
Nakaya commer-
(1959) cial ice sonic 5 to 50 12.7
Nakaya superim-
(1959) posed ice sonic 5 to 50 15.5
Nakaya ice with
(1959) elongated sonic 5 to 50 15.9
bubbles
Nakaya ice with
(1959) small sonic 5 to 50 18.7
bubbles
Hi ashi A. commer- hollow 1.9 and
1959 cial ice cyllinders 10.8 57.8
under hy-
drostatic
pressure

 

 

 

 

 

“Snow does not have steady flow; values are based on
issumption that equilibrium flow has been reached.

57

(1957) did not disprove Glen's (1955) value, as he was
attempting to relate the activation energy of snow to ice.

For snow AH varies from l4k cal/mole to 27k cal/mole.
Higashi's (1959) we“ cal/mole is the highest value reported.

He bases this value upon three tests.
Nakaya (1959), by observing the damping in oscillo-

grams of ice, has developed a technique of measuring the

viscosity of ice. Viscosity can be related to shear flow,
which in turn serves as a basis to compute the activation

energy. He finds that different types of ice do not have a
large variation in their activation energies, but their
respective Young's moduli vary over a large range. Jellinek

and Brill's (1956) A H agrees well with the results of

Nakaya (1959).
Referring to Higashi's (1959) statement that the

value of n in equation 5a should increase with stress, and

to equation 7, it is interesting to note that the activation
energy of ice may be a function of stress. If this were the

case, the behavior observed by Griggs and Coles (1954), Glen

(1955) (as interpreted by Higashi, 1959), and Higashi (1959),
could be explained. Both Higashi (1959) and Glen (1955)
determined the activation energy of ice from experiments

1Higashi's results are based on three samples, each

at 100 psi confining pressure. The straight line used to
determine A H is the "average" of these samples. Assuming
one sample was incorrect, activation energies could be
calculated ranging from 4 k cal/mole to 124 k cal/mole.

 

 

58

conducted at relatively large stress levels; their results

are canparable. The values of A H determined at low stress
levels (Jellinek and Brill, 1956) are comparable to the

values based on the sonic technique (Nakaya, 1959).

Equations 5a and.6 should be related in some fashion,
since they both represent minimum creep rate. Figure 9 shows
the activation energies found by the investigators of ice,
compared to the stress level for which each value was ob-

Nakaya's (1959) sonic tests were assumed to be

tained.
representative of the lower bound (creep under no load). From

this figure A H appears to be based on some function of stress.
Since this value will be an exponent in Eq. 7, and since both

Eq. 7 and Eq. 5a represent é , Higashi's conclusion concern-

ing equation 5a is supported.

Effect of Confiningjressure on Creep
Very little work has been performed on ice under

Rigsby's (1958) experiments are the most nota-

confinement.
By applying pressure to a single crystal of

ble to date.
ice under pure shear he was able to study the deformation

rates under varying confining pressures. Results of his

experiments (Figure 10a) show that at a constant tanperature,

the rate of deformation increased with the application of

pressure. His apparatus was capable of immense confining
His conclusions

pressures (1 atmosphere to 506 atmospheres).

are based on the results of four specimens.

 

 

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40

Realizing that confining pressure has an effect upon
the melting point of ice, he became curious as to the effect

of pressure upon deformation rate when the difference in

temperature between the sample and melting point of the sam-
ple was held constant (Figure 10b). His conclusions are
. . . that the shear strain rate for plastic deformation

of single ice crystals deforming by gliding on the basal
glide planes is practically independent of hydrostatic
pressure when the difference between ice temperature and

melting point is maintained at a constant interval.

Viscoelastic Properties of Ice
Viscoelastic deformation is the result of the com-

bined action of a material's elastic and viscous elements
when subjected to applied forces (Jastrzebski, 1959). Since
real materials are neither ideal liquids (viscous) nor ideal

solids (elastic), the last decade has seen increased accept-

ance of viscoelastic models to represent the behavior of
In their simplest form, they are com-

engineering materials .

posed of springs representing the elastic elements and dash-
These elements are

pots representing the viscous elements.
combined into various series and parallel configurations to

produce mathematical expressions for stress-strain-time

relationships which may suit a given material under study
The simplest models are those

(Secor and Monismith, 1961).
Their configurations are

of Voigt, Maxwell, and Burger.

shown in Figure 11a.
The Maxwell model is composed of a spring and a

dashpot in series, having an elastic element, E1, and a

 

 

 

 

 

  

 

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viscous element, 7 1. The Voigt model is composed of a

spring and dashpot in parallel, having elements of E2 and

77 2, respectively. The Burgers model is a combination of

the Maxwell model and the Voigt model. Where 0' is the stress,

6 the strain, and t the time, the behavior of the Voigt model
can be shown to be

0'(t)==E2 €(t)+>72 53—53%!)- .

The differential equation for the Maxwell model is

’21 1

A very important step in the understanding of visco-
elastic models was taken by Boltzman in 1878, in which he
related the behavior of a material to its past history.
According to this theory, the strain at any time due to a
variable load history is given by the sum of the strain con-
tributions from each past instant (Finnie and Heller, 1959).
This principle of superposition allows the behavior of the

Burgers model to be represented by the sum of the behavior

of a Maxwell model and a Voigt model. The mathematical

expression for strain of a Burgers model is as follows

6=g+£l+£

-t T '
E1 ’71 E2 (l-e / rot) . . (s)

The retardation time, Trot: can be closely approximated by
its relationships to the elements of the Voigt model, ’7 2,

 

 

43

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44

E2, and Poisson's ratio ,0, (Jellinek and Brill, 1956, as

_ Tret

’7 2/E2 " m)

 

The attempt to give physical significance to the
various elements of a viscoelastic model has been difficult

in the past. However, Goldman (1957) does present an idealized

Burgers model, assigning specific behavior to the several

elements of the model for an amorphous polymer (Figure 11b) .

His three categories are as follows:

1. A rapid, reversible elastic response (characterized
by a high modulus)-

2. Configurational elasticity associated with the straight-
ening out and orientation of the polymer chains. This
response is characterized by a low modulus of elasticity
and is time-dependent, being retarded by the internal

viscosity of the medium.
5. The irrecoverable slippage of entire polymer chains

past each other. This is the true viscous flow, and is
characterized by the melt viscosity of the polymer, the

feature persisting at higher temperatures.

These categories are dependent primarily upon the

 

chemical composition, molecular architecture, and temperature

of the medium. Viscoelastic materials are usually so complex

that no simple model will determine their behavior, and the
models must be modified by adding many Voigt models in series

or parallel.
Since the degree of complexity of observed behavior

of viscoelastic materials is astronomic, in the last few years

investigators have been in search of a series of simple tests

45

which will provide sufficient information to determine the
kind of model to represent the behavior and the magnitudes
of the elements in these various models.

The tests commonly used today are creep under constant
load, relaxation at a given strain, creep followed by removal
of the load, stress-strain tests with a constant-rate-of-
stress-application, and determination of Young's modulus by

resonant frequency. Several of these tests are described in

 

the following paragraphs, which explain how previous investi-
gators have used them to evaluate the elements of the models.

Nakaya (1959) assumed a Maxwell model for his study
in which the dashpot's coefficient of viscosity was deter-
mined from the damping of the vibrations produced in the test
specimens. The viscosity, he observed, increased in an
exponential form.with a decrease in temperature.

Jellinek and Brill (1956) successfully used a Burgers
model to predict the creep and relaxation of ice under low

stress conditions. Spring E1 should represent the instan-

 

taneous elastic response of ice (Goldman, 1957) (see Figure

12 and 15c). Jellinek and Brill (1956) determined the value
of this spring from the instantaneous deformation of a poly-
crystalline specimen just after the load has been applied

(see Figure 15). Their value for E1 was 814,000 psil, lower
than the value for an ice of comparable density (1,060,000 psi)

 

1The value used by Jellinek and Brill (1956) represents
the average of values ranging from 520,000 psi to 1,120,000 psi.

46

 

 

 

 

 

 

 

 

R
i
V)
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Burgers Mode/t/T l 060’ #/#%
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’(t‘to)
(_e A“) + Oat-to) I ' I /fi,
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47

as given by Nakaya, (1959).

The value of E1 may also be obtained from the re-
covered strain observed when the load is suddenly removed
from a creeping specimen. The instantaneous recovery strain
(the vertical straight line in Fig. 15b) has a magnitude of

0'/E1. Jellinek and Brill (1955) found that their values
of E1 determined in this manner agreed reasonably with those
found by the method described in the proceeding paragraph.

The bulk flow laws of ice which consider time as a

 

parameter represent viscoelastic flow. The viscous elements,
77 1 and ’72, of a Burgers model are time dependent. 7] 1 is
best calculated from a relaxation curve as shown in Figure
15c. 772 is best calculated from the slope of the "steady
state flow," the minimum creep rate (Jellinek and Brill,
1956; Finnie and Heller, 1959; Secor and Monismith, 1961).
Values of the viscous elements in 9. Burgers model
can be checked by comparing the measured strains in the pri-
mary creep stage (transition zone) to the strains predicted
by the Burgers model in primary creep (Jellinek and Brill,

1956) . The mathematical expression for strain of a Burgers

 

model in primary creep is

6 = J/Ez (l-e- At/Tret) + %£ .
1

Both ’7 1 and 7 2 are distorting in prMary creep (see Fig. 15c).

48

The elements of the Voigt model were determined from
the curved portion of the creep curve in Jellinek and Brill's
(1956) analysis. There was extremely good agreement between
the viscous elements found in this manner and those found
from the deformation recovery curves observed after the load
had been removed from creeping specimens.

The value of spring E2 may be determined either from
a creep curve or a relaxation curve. The strain found by
projecting the slope of the secondary creep stage back to the
initial deformation axis will allow a strain to be extra-
polated which is CT/Ez in magnitude. From a relaxation curve
this same value of strain, Cr/Ez, can be taken from the curve
as shown in Figure 12c. The elastic element E2, determined
by Jellinek and Brill (1956) apparently increases in magnitude
as the temperature decreases.

Density of ice has a pronounced effect upon the
viscous elements 771 and 72 (Nakaya, 1959). An ice having

1 will produce a creep curve

a density less than pure ice
showing higher rates of creep and larger deformations, i.e.,
its Burgers model elements would have smaller viscous
elements.

The viscous element 771, can be related to the acti-

vation energy as follows (Jellinek and Brill, 1956):-

-AH/RT
7 1 = const e

 

1Low density ice usually has a large degree of trapped
air bubbles in it.

49

It should be reemphasized that Jellinek and Brill's
(1956) analysis is based on a linear stress-strain-ttme'
relationship, i.e. Boltzman's principle of superposition is
valid. Recent investigations into the viscoelastic behavior
of textile fibers (Finnie and Heller, 1959) show that a
basic stress-strain relationship, such as Eq. 8, can'be used
even for non-linear responses. In this case the stress must
be raised to a power larger than one.

Most engineering materials show deviations from
linearity both in the elastic and viscous range, and their

behavior can be better approximated by the equation:

0‘
6t = gi- A 0.(1-e-qt) + Bant e e e e (9)

Where A, B, q, n, and d are constant characteristics for
each.material under specific conditions and must be deter-
mined from experiments. It should be noted that these
constants, although they have no real rheological meaning,
correspond to the rheological coefficients in the equation
for the behavior of the Burgers model. Thus A becomes
related to l/E2, q to l/Tret’ and B to 1/71. o<and n are
introduced to account for the deviations from linearity
(Van Vlack, 1959).

The intuitive feeling for viscoelastic behavior
obtained from a simple model is one of its chief attributes.
A schematic model representing non-linear behavior would be

composed of many Voigt and Maxwell models hooked in series

50

and/or parallel. Such a diagram would be so complex that
it would lose its ability to give the investigator this
"clairvoyant" sensation he so desires.

An understanding of the mechanical properties of ice
is necessary before satisfactory methods for the analysis of
a frozen soil can be formulated. As mentioned in the intro-
duction, ice within a frozen soil will be subjected to both
high stress and confining pressure, thus the mechanical
properties of polycrystalline ice under these conditions
should provide a clue to the rheological behavior of frozen
soil. In this study, creep experiments and constant rate
of strain experiments have been used to provide some answers

to the behavior of polycrystalline ice.

CHAPTER III
APPARATUS AND PROCEDURES

The water used in this experhment was distilled and
deionized. The water was checked for the amount of dis-
solved salts with an RD-D4 Solu-Bridge, which indicated that
on the average only 1.5 parts per million of dissolved salt
were present. All glassware that came in contact with the
water was first thoroughly cleaned with a sulfuric acid
sodium dichromate cleaning solution. Before using the water
it was boiled for at least 1 1/2 hours in a flask to drive
off dissolved air. One liter was sufficient to fill the
mold.

An aluminum mold similar to that used by Leonards
and Andersland (1960) was used to form the ice specimens
(see Figure 15). The mold was coated with a thin film of
Dow-Corning high-vacuum silicon st0p-cock grease to prevent
adhesion between the mold and ice. To prevent leakage be-
tween the mold and its baseplate, each was coated with
grease.

The water and the mold were both at room temperature
during the filling of the mold. To prevent small air bubbles
from mixing with the water, a 100 ml pipette was used to take
the water from the flask and place it in the ice mold. The

51

52

".1 Ash‘lmv‘ig" «-

 

Disassembled Aluminum Ice Mold, Specimens,

and Loading Paraphernalia

Figure 15

53

filled mold was then placed in a large refrigerator box which

maintained a temperature of -180 1 1° C, and the water was
allowed to supercool. The samples were frozen at -4° C 1 1/2°C
by seeding them with a crystal of distilled deionized ice.
Nucleation proceeded through the sample at a moderate rate,

and during the following growth the samples were not disturbed.

After 12 or more hours the ice mold was taken to another refrig-

erator box at -15° C. The mold was kept covered with aluminum

foil to prevent evaporation.
The top of the cylindrical samples were trimmed at

an age of two or more days with a single-edged razor blade
and allowed to remain in the mold until a few days before

testing. Any nicks accidentally cut into the tops of the

cylinders during trimming were filled with a drop of water

and retrimmed.
Before the ice specimens were placed in membranes,

they were visually inspected for cracks, if such were ob-
Bad cracks were uncommon.

the specimen was discarded.
A

served,
Specimens prepared in this manner were slightly cloudy.
few small spherical air bubbles could be seen in the samples.
Their density did not differ measurably from that of clear
ice (0.917 gin/ems). Four batches of ice specimens were

made. The last batch made had the lowest density, 0.915 gm/

ems. Specimens from this batch appeared to have more air
bubbles trapped in them than in previously prepared samples.

Since Jellinek's (1957) observations that adhesion to

loading plates and ice creates stress components in a lateral

54

direction, a friction reducer was used between the loading

cap and ice specimen. Leonards and Andersland (1960) used

thin hard rubber discs to reduce friction, while Butkovich

(1954) used chamois or sponge rubber.
The friction reducer selected consists of a three-

layered "sandwich," the tOp and bottom of which are thin

polyethylene plastic sheets and the middle layer is a per-

forated sheet of aluminum foil (Serata, 1961). Holding

these pieces together is a viscous mixture of stop-cock

grease and finely powdered graphite. Perforations in the

aluminum foil (quarter inch holes made with a paper punch)

provide small reservoirs of grease and graphite, preventing

the unit from working as one piece. Serata (1961) found

that these friction reducers work extremely'well even under

high pressures (9,000 psi). The friction reducers are pre-

compressed to drive out surplus grease by placing them

between two metal plates and compressing them with a Uni-

versal testing machine. These "sandwiches" may be used

several thes.
Since only a small amount of ethylene glycol (even

vapor) is sufficient to greatly alter the strength character-

istics of ice, it is imperative that the ice samples are

isolated from the cooling bath. Furthermore, evaporation of

the ice can be quite pronounced if the samples are not iso-

lated from the air (Glen, 1955). Glen (1955) tested his

55

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56

creep specimens by immersing them in a paraffin bath. Speci-

mens used in these tests were not exposed to the atmosphere

more than a few minutes.
Figure 15 shows schematically how the specimens were

Jacketed in membranes, how the loading caps and friction
reducers were placed, and how rubber bands were staggered

so that leakage between the membranes and the specimen was

impossible. A hole was provided in the top loading cap to

seat a removable steel piston cap.

A Tinus Olson Universal Testing Machine with a load

holding device was used for all tests conducted. The main

hydraulic system may be diverted through a small needle
valve, which allows more accurate control on the hydraulic

system of the loading head, hence control on the movement of

the loading head. It was possible to pre-set this valve,

causing the moving head to move at a prescribed rate. Setting

this rate immediately before a test was conducted (allowing

the hydraulic system to come to‘ equilibrium) insured a given
rate of deformation of the loading head. It was found that

the resistance given by the test sample was not sufficient

to alter the constant movement of the loading head. This

technique was used to produce the data for the constant-rate-

of-s train tests.
For the constant-rate-of-strain experiments loads

were read from the load dial at given intervals of time by

marking the position of the indicator on the glass dial with

57

a grease pencil. The deformation reading at the end of the

test was compared to the total time over which the test was

conducted. It was found that the predicted deformation and

actual deformation agreed within 1 0.0005 inches for the

total time of test.
Also connected to this needle valve is the load

holding device. A desired load may be "fixed" on the load-

ing dial by locking a pointer arm at the desired load. When

the moving load-indicator arm touches the fixed pointer a

circuit is completed and the needle valve closes, causing a

drop in hydraulic pressure and a reduction in load. When

the moving indicator arm is no longer in contact with the

fixed arm, the circuit is again open, and hydraulic pressure

is re-applied to the loading head.
This technique was used to obtain the creep data.

It was observed that the load varied, on the average of less

than _'I_-_ 2 pounds. In the case of creep at a very high load

level, it was found that the load was hard to hold as it had

a tendency to drop off. A variation of _4_-_ 5 lbs. was observed.

For this large a variation it was assumed that the sample had

failed.
Deformation of the sample was measured relative to
the loading head as compared to the base of the testing

machine. Measurements were read to the nearest 0.0005 inch.

Time was measured to the nearest second with a Lab-chron

Timer .

58

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59

A triaxial cell was used as a test cell. It was
modified in order that a thermocouple could be passed through
the cell and attached to the surface of the sample. This was
accomplished by tapping a hole in the metal cell tOp and
inserting a brass fitting so sealed that it would not leak
when the cell was under pressure. Figure 15 shows how the
test cell was placed in the testing equipment. To place the
specimen in the cell, a technique was develOped in the course
of the experimentation which exposed the frozen sample to a
minimum.of warming. The detailed procedure is outlined in
the appendix.

The schematic diagram of the pressure system used is
shown in Figure 16. The cell was filled by means of the
pump on the temperature bath through line HAG, with valve 0
closed. Stop-cock N was open, allowing the air in the cell
to escape. Air pressure was used to supply confining pres-
sure to the sample through line CG with.valve A closed. The
pressure was controlled by the double valve, D, and measured
with the gauge E. The cell could be drained easily by placing
it in the coolant bath, opening valves A and N'and lifting
the cell tOp from the base plate. When the fluid was under
pressure it flowed from line GAH without any additional pres-
sure.

Ethelyne glycol containing 50 per cent water by vol-
ume was used as a coolant fluid. A pump supplied the coolant
to the circulating tank which held the test cell. The tank,

60

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
 

 

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61

made from a standard five-gallon can, was constructed so
that the coolant entered at the bottom and returned to the
bath from a line at the top of the tank. This supply and
return of coolant was continuous throughout the period of
testing (see Figure 15) and maintained the cell temperature
to i o.os° c. The cell was isolated from the bottom of the
tank by a circulation plate which allowed the coolant to
flow beneath the cell as well as around it. This minimized
temperature change by heat conduction through the base.

Since moisture condensed on the outside of the tank,
a spill pan was placed under the tank to catch the condensa-
tion. This spill pan also caught any coolant that might
flow over the sides of the tank when the test cell was placed
in the tank, and served as a safety device should the line
returning to the bath become ineperative. To prevent errors
in deformation readings the test tank was supported by a
circular plate placed between the tank and the spill pan.

The low temperature bath.was controlled by a mercury
thermostat submerged in the bath. The temperature differ~
ence between the tank and bath was approximately one degree
centigrade. Because the temperature did not remain constant
*within the room it was necessary to change the setting of
the thermostat periodically.

The temperature of the sample was obtained by use of
£1 cOpper-constantan thermocouple (Scott, 1941), and a Leeds

sand Northrup potentiometer, Model Kp2. A bath of distilled

62

deionized, melting ice was used as a reference temperature.
A standard cell (1.0915 volts) and a Leeds and Northrup
Galvanometer aided in.measurement. These instruments were
powered by a standard car battery. The thermocouple ar-
rangement was capable of reading to the nearest 0.02° o.
The temperature measuring equipment is shown in Figure 17.

Based on the temperatures measured on the surface
of specimens, it was found that the cooling system was quite
efficient. In the circulating tank the temperature of the
entering fluid was 1/20 C colder than the temperature of
the coolant returning to the source (temperature bath). The
coolant in the triaxial cell was not moving and protected
the specimens from any temperature variation of the circulat-
ing tank. During the time taken to transport the specimens
from the cold storage and to mount them on the triaxial cell,
the specimens were exposed to room temperature for a short
period of time. During the filling of the test cell the
temperature of most specimens was ~90 1.20 C. All experi-
ments were conducted at -4° C1_0.05o C. Several of the
first specimens used cooled to the test temperature after
having warmed to -5° 0 during mounting.

It was found that the frozen samples would reach the
test temperature within two and one-half hours. Providing
the circulating tank was at the correct temperature level,
the temperature at the surface of the specimen remained

nearly constant (30.050 C) indicating thermal equilibrium

63

  
      
    
    

  

 

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SPQCim'

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-’

Temperature Measuring Equipment

Figure 17

64

had been reached. To make sure that thermal equilibrium
had been reached the specimen and its surrounding coolant
were allowed twelve or more hours (overnight) in.which to
reach equilibrium with each other. A few duplicate rate
of strain specimens were exceptions as they were tested
four hours after being placed in the test cell. No notice-
able difference was observed in the results of these
specimens when they were compared to their duplicate cem-

panion.

CHAPTER IV
EXPERIMENTAL RESULTS AND DISCUSSIONS

Results of Constant Rate of Axial Deformation Experiments

Twenty cylinders of polycrystalline ice were tested
under constant rates of axial deformation until failure occur-
red. Four rates were used, approximately doubling the rate
for each series of experiments. The rates used are 0.54%,
0.95%, 1.76% and 5.53% expressed as per cent of axial deforma-
tion per minute. Confining pressures of 0 psi, 50 psi and 45
psi were used. All results are summarized in Figures 20a, b,
c, and d.

The temperature at the beginning of each experiment
was -4° C 30.050 C. The temperature was checked periodically
during the experiment and a trend for the sample to warm was
observed. This may be due to the thermal energy created as
the specimen deforms.

Experiments performed with the fastest rate of load-
ing required less than two minutes to produce failure of the
specimen. For this rate load readings were taken at two
second intervals with the help of a timekeeper.

Most of the constant-rate-of-strain specimens that
failed had a shattered appearance when taken from the test
cell (Butkovich, 1954). A typical failed specimen is sketched
in Figure 18.

65

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67

In all samples tested there was a time lag before
the load indicator responded to the deformation applied to
the sample. In soil testing this effect is eliminated when
the loading caps are properly seated on the specimen. Sev-
eral factors may influence the seating strain for the ice
samples. They are:

1) Compression of the friction reducers

2) Response of the lucite loading caps

5) Seating of the specimen on the sample mount

4) Seating of the loading caps on the specimen.

A typical response of ice to a constant rate of strain is
shown in Figure 19. The seating strain is defined as the
intercept strain found by prolonging the elastic portion of
the stress-strain curve (viz. line AB in Figure 20) to the
strain axis. This is shown as point C in Figure 19.

A comparison of the stress at which point A occurred
shows that, for a given strain rate (1.76% per minute), ten
of fourteen samples began their elastic response between 67
psi and 73 psi. The fourteen samples are composed of six
constant-rate-of-strain experiments and eight approach curves
for creep. The samples which did not fit into the previously
given range of stress began their elastic response at 48 psi,
80 psi, 106 psi and 118 psi. The seating strain, as previously
defined, varied from 0.0056 to 0.0081 inches/inch.)

Values of Young's modulus were determined from the

stress-strain curves in Figures 21a, b, c and d and are

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TABLE 3

SUMMARY OF CONSTANT-RATE-OF-STRAIN RESULTS

 

 

 

 

 

 

 

 

 

 

Sample 5,% gpsi minim/in2 5 p 6 ult Jult,psi
I - 1 0.95 0 19.62 .0185 .0260 456
I - 2 0.95 0 35.09 .0110 .0160 495

5 0.95 0 35.80 .0105 .0170 439
4 0.95 30 23.52 .0105 .0212 346
6 0.95 30 27.44 .0090 .0265’ 362
7 0.95 45 31.87 .0096 .0213 400
8 0.95 45 30.27 .0108 .0250 433
9 0.54 0 25.30 .0083 .228 400
10 0.54 45 22.79 .0093 .178 390
11 1.76 0 50.00 .0086 .0118 485
14 1.76 0 32.95 .0112 .0273 473
12 1.76 30 31.65 .0123 .0182 476
15 1.76 30 35.70 .0112 .0166 483
13 1.76 45 30.00 .0100 .0168 398
16 1.76 45 33.70 .0133 .0190 543
17 3.33 0 37.50 .0105 .0162 483
20 3.33 0 46.71 .0110 .0137 565
18 3.33 30 47.29 .0113~ .0168 530
19 3.33 45 53.12 .0095 .0140 580
21 3.33 45 65.22 .0096 .0113 685
6, Strain Rate in % per minute
03, Confining Pressure
E , Young's Modulus

Strain at Beginning of Plastic Response

Failure Strain

Failure Stress.

 

summarized in Table 3.

strain.

The strains are adjusted for seating

Values of Young's modulus determined in this manner

show a very definite trend, but vary as much as 20,000 psi

74

for a given strain rate. The average value for Young's modu-
lus increases as the strain rate increases. For example, the
value of E corresponding to a strain rate of 0.54% per min-
ute is about one-half the value of E corresponding to a strain
rate of 3.33% per minute. The magnitude of Young's modulus
found in this manner is considerably smaller than the value
found by previous investigators using sonic methods which
corresponds to the initial tangent modulus.

This behavior may be illustrated by the Burgers visco-
elastic model (Figure 12). Spring El represents the initial
tangential modulus. Deformations for strain rates less than
infinity will be influenced by the viscous elements §71 and
7'2. The slower the strain rate, the lower is the observed
Young's modulus. Changes in Young's modulus with the rate
of load application are usually attributed to viscous flow.
Figure 21 shows the variation of Young's modulus with the
strain rate. The average value of Young's modulus plotted
nearly linearly with the strain rate. The line drawn through
the average values of E may be represented by the equation

, E=19.5+9.2é........(10)

where E is the value of Young's modulus and A is the rate of
axial deformation expressed in per cent of height per minute.
Any effect of confining pressure upon the value ostOung's

modulus is obscured by the scatter of results. Since temper-

ature influences the value of Young's modulus (Nakaya, 1959),

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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the constant in Eq. 10 should increase with decreasing tem-
perature. 0n the average, tests conducted at lower rates of
strain showed less ultimate strength than those strained at
higher rates. This behavior is observed in many engineering
materials (steel is a good example) where impact tests indi-
cate a larger ultimate strength of the material than those
conducted by conventional tension or compression tests. By
comparing Figure 20a (strain rate 0.54%) to Figure 21d
(strain rate 5.55%), it can be seen that the modes of failure
are becoming more brittle.

The arithmetic average of the different ultimate
strengths, irregardless of confining pressure, increases
linearly with increasing strain rate as shown in Figure 22.
The ultimate strengths varied as much as 200 psi. Previous
investigators of compressive strength of ice show a similar
scatter of results (Butkovich, 1954; Leonards and Andersland,
1960). The magnitudes of compressive strength are in the
same range as reported by Butkovich, 1954. Leonards and
Andersland (1960) conducted experiments in a.manner quite
similar to that used in these experiments. They used a con-
stant-rate-of-strain of 2% per minute. The range of ultimate
strength obtained by extrapolation of their results is plotted
in Figure 22. Their different range may be due topa slightly
different technique used in sample preparation. They did not
seed the supercooled water, hence less uniform samples may

have been obtained.

77

 

    
  

 

 

 

 

 

 

 

   

 
  
  

  

 

 

 

 

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An empirical relationship suiting the arithmetic

average of the results shown in Figure 22 is

O'ult =ses+ 65.55 . . . . . . (11)

where Crult is in psi and éiis the strain rate in per cent
of axial deformation per minute. As with the constant in
Eq. 10, the constant in Eq. 11 should increase with decreas-
ing temperature.

Any relationship between confining pressure and the
magnitude of ultimate strength is lost in the scatter of
results. In the creep of single ice crystals confining
pressure appears to soften the ice crystal. Comparisons be-
tween ultimate shear strength and confining pressure are
often made using Mohr's circles. If the confining pressure
is large enough, ice can no longer exist as a solid. At
about 7,160 psi and -4° C, isothermal reversible compression
causes the ice to melt, with a subsequent reduction in shear
strength to zero. Liquid water is represented by a point on
the Mohr diagram. Mohr's failure envelope should slope down-
ward toward this point, indicating a negative angle of
internal friction, - 0 . Mohr's circles drawn for the results
of the constant-rate-of-strain experiments neither confirm
nor disprove this.

For the range of stresses encountered in frozen soils,
the friction angle for ice might well be approximated by zero

degrees. Figure 25 shows the Mohr's circles for a strain

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rate of 1.76 per cent per minute. The maximum shearing
strength is taken as one-half the diameter of the Mohr's
circle of stress. The Mohr's circles plotted represent
average results, the actual magnitude of maximum shear strength
for individual specimens is shown by symbols. The stress axis
is broken to show the liquidus point and the downward slope of
the failure envelope.
Results of Creep Experiments

Eight polycrystalline ice specimens were used to ob-
serve creep pr0perties under constant load. Compressive
loads of 200 lbs (150.7 psi), 500 lbs (196.2 psi), and 400 lbs
(261.7 psi) were used along with confining pressures of 0 and
45 psi. All results are presented in Figures 24a, b, and 0.
Good agreement was found between identical samples (Figure 25b)
tested under a load of 500 lbs. Although the curves do not
fit one upon the other, the rates of creep are in close agree-
ment.

All test loads were approached at a strain rate of
1.76% per minute. Load and deformation readings taken every
five seconds made it possible to determine the seating strain
for each specimen (values ranged from .0056 to .0081 in/in).
The seating strain was subtracted from the strains observed
during creep. This technique is similar to that used by
Butkovich and Landauer (1959) in which the "elastic" strain
was subtracted from following strain readings.

After the tests were completed, inspection showed that

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84

the majority of creep samples appeared to have failed by
bulging. Five of the samples deve10ped "step-like" surfaces
(Butkovich and Landauer, 1959), indicating preferential
gliding (Figure 18). In addition to bulging, specimens I-29
(400 lbs) and I-50 (400 lbs, 45 psi) appeared to have been
shattered internally, similar to the cracked appearance of
the constant-rate-of-strain specimens. Cracking sounds were
heard during the testing of these specimens (see Figure 240).
The relationship between time and strain did not
prove to be linear (Griggs and Coles, 1954; Jellinek and
Brill, 1956; Butkovich and Landauer, 1959; Higashi, 1959).
Equations 5a and 5b are unacceptable. However, on a log
strain log time plot the creep data becomes linear, showing
a minimum creep rate and a tertiary creep rate. Behavorial
tendencies observed from the creep curves shown in Figures
24a, b, and o are noted below.
Minimum Creep Rate Observations
l) The minimum creep rate increased as the stress
level increased. Previous investigators of
creeping ice have observed this phenomenon. The
value of the strain at the beginning of secondary
creep was not pr0portional to the stress level, a
necessary condition for a viscoelastic model
analysis (Jellinek and Brill, 1956). (See Figure
25a, a summary of Figures 24a, b, and c).

2) The minimum creep rate is increased by application

I: {all 5‘ .Ill‘lsliiswnl

 

85

of confining pressure in the majority of experi-
ments. This observation agrees with Rigsby's
(1958) observations that confining pressure pro-
duces a "softening" of the crystals' bonds. The
one exception to this behavior is specimen I-24,
creeping under 200 lbs and confined at 45 psi.
Since Rigsby's experiments were performed on
single ice crystals, no conclusion can be made
about the "softening" of polycrystalline ice, as
bonding between the individual ice crystals
appears to be the controlling factor (Glen and
Perutz, 1954; Steinemann, 1954; Glen, 1955, 1958;
Butkovich and Landauer, 1959).

Stress level plays a determinant role for the results
of the eight creep samples. Values of the :xponent r‘in
equation 1 are computed for the minimum creep rate. It in-
creases with both stress and confining pressure (see Table 4).
Equations 1 and 2 are not recommended. (Butkovich and Lan-
dauer, 1959).

The technique of variable isolation used by Jellinek
and Brill (1956) for equation 4 proved to be difficult to
use. For the 400 lb creep curves, values of strain at 1,000
and 2,000 seconds had to be extrapolated in order to plot the
stress-strain at constant time (see Figure 25b). Analysis of

creep specimens by this technique was not successful. The

value of n varied between 0.60 and 1.15. At only one time

86

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 4
SUMMARY OF CREEP RESULTS
' ”1 0'3
Sample Load ps 1 psi 6 m Eb E t *min ’- tart
1-25 200 150.7 0 .0095 .0129 .0265 .225 .721
I-25 500 196.2 0 .0109 .0140 .0265 .176 1.145
I-28 500 196.2 0 .0105 .0140 .0258 .179 1.125
I-29 400 261.7 0 .0109 .0150 .0290 .598 2.086
1-24 200. 150.7 45 .0129 .0162 .0275 .145 .677
I-26 500 196.2 45 .0175 .0212 .0265 .597 .802
I-27 500 196.2 45 .0145 .0182 .0278 .555 .952
1-50 400 261.7 45 .0098 .0140 .0260 .704 1.711
Ein’ strain at end of minimum creep rate
6 b’ strain at assumbed beginning of tertiary creep
€.t, strain where tertiary creep is completely
established
f'min! exponent of Eq. 1 for minimum.creep
rtert’ exponent of Eq. 1 for tertiary creep.

 

level, t= 250", did a linear relationship exist when Figure

25b was replotted as log stress versus log strain.

The cor-

responding value of n for t = 250" was 0.80.

Specimens I-29 and I-50 were taken from the last batch

of identical specimens having a slightly lower density than

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88

the other batches. These specimens should and did exhibit
faster creep rates. If the creep rates of these samples
were somewhat lower, the value of n within the limits given
in the preceeding paragraph would satisfy Eq. 4.

Using n = 0.80, the exponent of the time variable
m was determined using the previously discussed technique.
Jellinek and Brill (1956) show the relationship with all
their results plotting on one line. The creep results showed
different slopes for each sample.1' Three samples had com-
parable results for which.m was approximately 0.20. These
three samples, I-25, (200 lbs), I-27 (500 lbs) and I-28
(500 lbs) were not subjected to confining pressure. The
specimens subjected to confining pressure had larger expo-
nents for both.m and n (except sample I-24; 200 lbs, 45 psi).
No basic relationship could be found between the respective
magnitudes of the exponents for confined specimens and the
magnitudes of the unconfined specimens.

The minimum creep rate of ice under high stress and
confinement was non-linear to such a degree that representa-
tion of behavior by a viscoelastic model is untenable.
Higashi (1959) reached a similar conclusion in his study of
ice under hydrostatic pressure.

Griggs and Coles (1954) also observed the nonlinearity

of time and strain in creep specimens. Their empirical

 

1A similar plot using tertiary creep is shown in
Figure 27, using n = 1.0.

89

relationship for single ice crystals, Eq. 5, includes the
temperature parameter. Rigsby's (1958) findings suggest that
confining pressure is also a parameter. Using Eq. 4 as a
framework, both temperature and confining pressure can be
included in an empirical relationship compatible with observed

behavior, generally

where f (673) and f (To) mean function of confining pressure
and temperature below freezing, respectively.

The minimum creep rate data was programmed on the
Michigan State University digital computer, MISTIC, for equa-
tion 12. Values of m and n were allowed to vary from .6 to
2.0 and from .6 to 4.5, respectively. Various functions of

0"3 were tried, as follows:

es» fa? + 0‘s and “Tel
0.5 was used for the value of/x, Poisson's ratio (Jellinek
and Brill, 1956). The computer was asked to solve for the
constant a. Trends observed from the variability of the
constant value, a, indicate:
1) m will be less than 0.6, probably between 0.2
and 0.55.
2) n will probably be less than one. The value at
0.8 worked well.

 

The effect of usingl- 0'3 as a function can be seen
in Figure 26, lines A and A .

90

3) The function of 0‘3 which produced the best
results was +0’ . Higher powers of 0’3 may
produce better answers.

For the prediction of creep, an empirical relationship in
the form of Eq. 12 appears to be the most suitable.
Tertiary Creep_0bservations

l) The time taken to begin tertiary creep decreased
as the load level was increased.

2) Tertiary creep rate increases with the load.
Triaxial test results agree with results using
other methods. (Steinemann, 1954).

5) Tertiary creep rate was decreased by confining
the sample.

Values for the exponent r in Eq. 1 were computed to
show the influence of confinement and load level. They are
listed in Table 4.

The behavior of the tertiary creep rate can be
observed in Figure 26. Once again difficulties arose when
using the analysis technique presented by Jellinek and Brill
(1956). Values of n in Eq. 4 for tertiary creep could not
be obtained in this manner because time could not be isolated.

By assuming a value of n = l, the exponent of the
time variable, m, was computed from the slopes of the lines
in Figure 27. The straight lines representing the tertiary
creep rate became apparent when they were compared to a

fixed point extrapolated from each specimen's creep curve.

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The fixed point was the ”beginning" of tertiary creep.

The beginning of tertiary creep is assumed as the
intersection of the minimum.creep line with the tertiary
creep line, symbolized by 5b. Eb agreed well from test
to test, varying in values of strain from 1.4% to 2.1%, aver-
aging at 1.5%. The strain at the end of the minimum creep
stage, i.e., the point at which the creep curves start a
transition into tertiary creep, is designated as 6m. The
point at the end of the transition into tertiary creep is
designated as 6t (see Figure 250). Values of strain at
these points are listed in Table 4.

Values of m varied with the stress level as is shown
in Figure 27. The interesting observation is that the effect
of confinement is very nearly constant. The relationship

between m unconfined and m confined at 45 psi is as follows:

m
_ unconf

m45 psi "' W o o o c o o o o o s o (13)

The nonlinearity of m with stress level made the use of a
viscoelastic model to represent tertiary creep unsatisfactory.
Several observations can be made by comparing the
results of the constant rate of strain eXperiments to the
creep experiments as shown in Figure 28. For the constant-
rate-of-strain experiments, values of strain at the begin-
ning of the plastic response (end of the elastic response)
and the strain at ultimate load are listed in Table 5. They

are denoted by 6p and Eult respectively. The listed

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94

values have been corrected for seating strain, and were
obtained by interpolation of the stress-strain curves,
Figures 20a, b, c, and d.

The arithmetic average strain value for the begin-

ning of the plastic zone, 63 was 0.0108. The strain

p;
value for ten of the specimens was within :_.0008 of this
value. Sixteen of the samples were within 1_.0018 of the
average value. It appears, then, that in the neighborhood
of 0.0108 strain, the polycrystalline ice samples begin
another behavioral trend. This behavior was also noticed

in the creep specimens where the average strain value at

the end of the secondary creep stage, 6 m’ was 0.0115 (see
Figure 28). Creep strains were also computed by subtracting
the extrapolated seating strain. It appears that G p is
approximately equal to é'm.

This raises the question, how is tertiary creep
related to the ultimate strength of ice? A comparison of
these two phenomena might be made by considering the rela-
tive amounts of energy supplied to a specimen that would
produce these behavioral characteristics. Energy calcula-
tions would necessarily be based upon the theory of plasticity
(Hill, 1956), and they would involve measurements beyond the
capabilities of the apparatus and technique used..

This question can be answered, in part, by a compar-

ison of the strains occurring at the beginning of tertiary

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96

creep, E b’ the ultimate strain, E ult’ and the strain at
which tertiary creep has been completely established,€lt.
This comparison may be made from strain values listed in
Tables 5 and 4. The scatter of results is shown graphically
in Figure 28 which shows the observed behavior of a specimen
tested at a strain rate of 1.76%, i.e. the same strain rate
at which the creep samples were subjected to during appli-
cation of their respective loads.

The average strain at which tertiary creep is estab-
lished, 6 t' was observed to be larger than the average
ultimate strain, C'ult’ 63b was found to be slightly less

than 6 but considering the scatter of results, their

ult’
values are comparable to each other. All observations con-
cerning characteristic strains of polycrystalline ice can

be stated approximately by
m
€P=€m<€b""€u1t< €t'°"°°‘(14)

The magnitude of strain at the beginning of plasticity is
approximately equal to the strain at the end of minimum
creep. These strain values are less than strain at the
"beginning“ of tertiary creep, and the ultimate strain, whose
magnitudes are comparable. The magnitude of ultimate strain
is less than the strain at which tertiary creep has been com-
pletely established.

Previous investigators have hinted that tertiary creep

97

begins after the bonds have been broken between the individ-
ual crystals of polycrystalline ice. It seems logical to
assume that tertiary creep begins when the ultimate strength
of the material has been surpassed. This hypothesis cannot
be entirely supported by the results, but based on the ob-
served strains it appears to be verified.

Observations made during the creep of the specimens
under a 400 1b load imply that the ultimate strength of
these two specimens had been reached during the transition
into tertiary creep. A slight cracking sound was heard
similar to that which was sometimes heard as a constant-
rate-of-strain specimens reached their ultimate strength. In
order to keep the load at 400 pounds, additional hydraulic
pressure had to be applied to the loading head of the test-
ing machine through its needle valve.

The hypothesis is further supported by observations
made on confined tertiary creep. Assuming the bonds between
the individual crystals had been broken, gliding would pro-
ceed almost entirely by movements between the crystals. The
tertiary creep rate would represent a resistance created by
friction between their surfaces. The effect of confining
pressure would create additional friction, retarding the
movement of the crystals, which in turn would be observed
through a lower tertiary creep rate. This effect of confine-
ment can be observed from Eq. 15. In addition, the reduction

in tertiary creep by a confining pressure of 45 psi was very

98

nearly a constant, indicating that a comparable amount of
resistance to flow was created, irregardless of the load
used for creep. This behavior supports the concept that
the intercrystalline bonds have been disrupted.

The nonlinearity of observed strains with stress in
both secondary and tertiary creep support Higashi's (1959)
conclusions concerning the exponent n en equation 4. n’s
value increases as the creep load increases. The previous
implication that the activation energy of ice may be depend-

ent upon stress level is also supported by this non-linearity.

CHAPTER V

CONCLUSIONS

From the observed results of the experiments con-

ducted on duplicate samples of polycrystalline ice at -4° C

and for the techniques used in this study the following

general observations are made.

1)

2)

3)

Duplication of tests is a necessity, as even
carefully duplicated polycrystalline samples
produced a large scatter of results.

The value of Young's modulus in compression is
highly dependent upon the rate of strain appli-
cation. An approximate relationship between

Young's modulus and the strain rate would be:

E 19.5+ 9.26 , 0.5%<é< 3.33%

where E is in kips per square inch and the strain
rate, é , is expressed in per cent of height per
minute.

The ultimate strength of ice increases with in-
creased rate of strain application. An approxi-
mate relationship between ultimate strength and

strain rate would be
fult = 565 + 65.5 ; o.5%<€ < 3.33%

99

4)

5)

6)

'7)

8)

100

where the ultimate strength is in pounds per
square inch and the strain rate is as previously
defined.

The use of a viscoelastic model or simple bulk
flow formula to represent creep of ice under high
stress appears to be untenable except for approx-
imate relationships.

The effect of confining pressure upon the creep
of ice under high stress appears to increase the
minimum creep rate and to decrease the tertiary
creep rate. Further studies are needed to deter-
mine the actual influence of confining pressure.
The total strain values at the end of the elastic
response for the constant-rate-of-strain experi-
ments agrees well with the total strain at the
end of the minimum creep state for the creep
results.

Using a similar comparison of strain values, the
ultimate strain at failure in the constant-rate-
of-strain experinents compares with the assumed
"beginning" of the tertiary creep curve.

Based on strain measurements and supported by
conclusion number five, it is likely that constant
tertiary creep begins after the ultimate strain of

the polycrystalline samples has been exceeded.

RECOMMENDATIONS FOR FURTHER RESEARCH

Further research is needed in several areas related
to the mechanical properties of ice. Specific problems are
briefly described below:

1. Study the effect of large confining pressures at
constant temperatures to determine the shape of Mohr's rup-
ture envelope. The angle of internal friction for ice should
be investigated when the difference between the melting point
and test temperature is held constant similar to the technique
used by Rigsby (1958).

2. Study the relationship between activation energy
and more conventional bulk flow laws. There is an indication
that H is dependent upon stress.

5. Study the relationship between the creep curves
of a frozen fine grained soil and polycrystalline ice. A
further comparison with duplicate unfrozen soil specimens
may provide answers to some of the more fundamental questions
about frozen soil mechanics.

4. Determine the Universal Stress-Strain Curve for
ice (Hill, 1956) in order that comparisons can be made between
constant-rate-of-strain experiments and constantérate-of-
stress-application experiments. Once it is determined, the
tensile strength of ice (Jellinek, 1957) might be compared to

the compressive strength of ice.

101

BIBLIOGRAPHY

Bjerrus, Niels. "Structure and Properties of Ice." Science
Vol. 115, London, England, 1952, pp. 585-590. _._._....._.._

Butkovich, T.R. Ultimate Strength of Ice, U.S. Army Snow,
Ice, and Permafrost Research Establishment, Corps of
Engineers, hereinafter referred to as SIPRE, Research
Paper 11, Wilmette, Illinois, December, 1954.

 

, and J.K. Landauer. The Flow Law for Ice. SIPRE
Research Report 56, August, 1959.

Chalmers, Bruce. The Growth of Ice in supercooled Water.
(Edgar MarburgILecture, 64th Annual Meeting of the Amer.
Soc. for Testing and Materials, October, 1961, Phila-
delphia, Pa.)

 

Finnie, I. and W.R. Heller. Creep_of Engineering Materials.
New York, New York: McGraw-Hill Book 00., Inc., 1959.

 

Glen, J.W. "The Creep of Polycrystalline Ice." Proceedings
of the Royal Society of London, Vol. 228,1175, Series A,
London, England: Royal Society, March, 1955, pp. 519-558.

. "The Mechanical Properties of Ice.” Advances in
Physics, Vol. 7, London, England: Taylor and Francis
Ltd., 1958, pp. 254-258.

 

, and M.F. Perutz. "The Growth and Deformation of
Ice Crystals." Journal of Glaciology, Vol. 2, Cambridge,
England: British Glaciological Society, 1954, pp. 597 to
405.

 

Goldman, an. "Structure and Viscoelastic Pr0perties of
Amorphous." The Science of Engineering_Materials,
New'York, New York: John Wiley & Sons, Inc., 1957,
pp. 472-479.

Griggs, D.T. and N.E. Coles. Creep of Single Crystals of
Ice. SIPRE Report 11, 1954.

 

Halvorsen, L.K. Detennination of the Modulus of Elasticity
9f Artificial Snow-Ice in Flexure. SIPRE ResearCh Report
51, February, 1959.

 

 

102

103

Higashi, A; Plastic Deformation of Hollow Ice Cylinders Undgg
HydrostatiE Pressure. SIPRE Research Report 51, July, 1959.

 

Hill, R. The Mathematical Theory of Plasticit . Oxford,
England: ‘OxFSrd at the Clarendon Press, 1 56.

 

Jackson, KJA. and B. Chalmers. Stud of Ice Formation in
Soils. (Artie Construction and rost Effects Laboratory,
for Office of the Chief of Engineers, Airfields Branch,
Engineering Division, Military Construction, Technical
Report No. 65, November, 1957.) Boston, Massachusetts:
U.S. Army Corps of Engineers, 1957.

Jastrzebski, Z.D. Nature and Properties of E ineeri
Materials. NewEYork,‘NewfiYork: JEhn Wiley 5 Sons, Inc.,
1959.

 

 

Jellinek, H.H.G. Tensile Stre th Properties of_Ice Adhering
to Stainless Steel. SIPRE Research Report 25, january,

1857 .

and R. Brill. "Viscoelastic Properties of Ice.”
journal of Applied Physics, Bul. 27; 10, New York, New
Yerk: Amer. Inst. of Physics, October, 1956.

Kamb, W.B. "The Glide Direction in Ice.” Journal of Gla-
ciolo , Vol. 5, Cambridge, England:. BritiEh Glaci-
ological Society, October, 1961, 1097-1106.

Landauer, J.K. Creep of Snow Under Combined Stress. SIPRE
Research Report 41, Disamber;“19§7.

 

. Growigg of Large Single Crystals of Ice. SIPRE
Report 48, July, 1958.

 

Leonards, G.A. and 0.8. Andersland. "The Clay Water System
and Shearing Resistance of Clays." Research Conference
on the Shearing Strength of Cohesive Soils, Soil
Mechanics andiFoundations‘Division, Amer. Soc. of Civil
Engineers, University of Colorado, Boulder, Colo., June,
1960, pp. 795-817.

Lovell, C.W., Jr. Temperature Effects on Phase Composition
and Stre th ofIPartiaII¥:Frozen Soil. *HighwayIReseafEh
Board BuIIeEIn 168, Waéh.ngton, D.C., 1957.

Martin, R. Torrence. "Rhythmic Ice Banding in Soil! Frost

Effects in Soils and on Pavomgpt Surfaces, Highway
Research Board Bulletin 218, Washington,7D.C., 1959.

 

Nakaya, U. Mechanical Properties of Single Crystals of Ice.
SIPRE Researéh.Report 28, 06t0ber,‘1958. I

104

. Viscoelastic Properties of Snow and Ice in the
Greenland Ice Cap. SIPRE Research Report 46, May, 1959.

 

Richards, E.W. Engineering Materials Science. San Francisco,
California: “Wadsworth Publishifig 00., Inc., 1961.

 

Rigsby, G.P. ”Effect of Hydrostatic Pressure on Shear De-
formation of Single Ice Crystals." Journal of Glaci-
olo , V01. 5, Cambridge, England: British Glaciological
500., October, 1958, pp. 275-278. (Originally published
as SIPRE Research Report 52.)

Scott, R.B. "The Calibration of Thermocouples at Low Tempera-
tures." TemperatureL_Its Measurement and Control in
Science and Industry. (Amer. Inst. of—Physics andINational
Bureau of Standards, National Research Council) New York,
New'York: Reinhold Publishing Corp., 1941, pp. 206-218.

 

Secor, K.E. and C.L. Monismith. "Analysis of Triaxial Test
Data on Asphalt Concrete Using Viscoelastic Principles.”
Proceedings of the Highway Research Board, Vol. 40, The
Highway Research Board, Washington, D.C., January, 1961,
pp 0 295-314 0

Serata, S. (Asst. Professor of Sanitary Engineering, Michigan
State University), Personal communication, December, 1961.

Skempton, A.w. ”Effective Stress in Soils, Concrete and
Rocks." Pore Pressure and Suction in Soils. London,
England: _Bu1terworths, Inc., I961.

Steinemann, Samuel. "Results of Preliminary Experiments on
the Plasticity of Ice Crystals.” Journal of Glaciolo ,
Vol. 2, Cambridge, England: BritisE GIacIoIogIcaI 300.,
1954, pp. 404-412.

Stephens, R.W.B. ”The Mechanical Properties of Ice. The
Elastic Constants and Mechanical Relaxation of Single
Crystal Ice.” Advances in Ph sics, Vol. 7, London,
England: Taylor and Francis Ltd., 1958, pp. 266-275.

United States Army Corps of Engineers, The Frost Effects
Laboratory. Investigatiog of Description, Classification,

and Stre th.Pro erties of Frozen SoiIs. SIPRE Repoft 8
Fiscal Year 1951, June, 1952.

Van Vlack, L.H. Elements of Materials Science. ‘Reading,
Mass.: Addison-Wesley Publishing Co., Inc., 1959.

 

APPENDIX‘A

Detailed Procedures

106

Mounting A Frpsen Sample In The Triaxial Cell

A Leonards-Parnell Triaxial Cell was used as a test
cell. The top and cylindrical sides are built as one piece;
the base is separate.

To place a sample in the cell, a technique was devel-
Oped which exposed the frozen sample to a.minimum of warming.
Samples were carried from cold storage (-150 C) to the test
cell in a pro-chilled brass membrane stretcher and immediately
mounted in the test cell. The test cell was submerged in the
coolant bath, which was colder than the test temperature. The
sample mount on the test cell base was not entirely submerged.
The sample was placed on this projection and the surplus of
its two membranes rolled over the mount. A heavy natural
rubber'membrane was then placed over the sample and mount,
and the thermocouple inserted between this new membrane and
the sample.

The test cell was then taken from the bath and placed
in its proper position on the cell base. The piston was
lowered and seated on the piston cap, after which the top of
the cell was secured to the base plate. The piston was then
looked in place so that the sample would not become unseated.
during moving. A j

The test cell was allowed to fill while submerged in
the coolant bath. It was then placed in the filled and cir-
culating test tank. Pressure was applied to the system

107

either by opening the cell stopcock, (0 psi), or applying
pressure by means of the hydraulic system, (30 psi and
45 psi).
Calculation of Stress

In the constant rate of strain experiments, the
observed deformations were converted into unit axial strain.
As the sample was compressed, the average cross-sectional
area became larger. Based on the theory of plasticity,
(Hill, 1956), this enlargement was accounted for in the

calculation of stress by:

€7=LP
I'1

when the corrected area, A1, is given by

A1 3A0
I=3train

A is the cross-sectional area at the beginning of the

experiment. In the creep experiments no adjustment was
made to account for the increasing cross-sectional area.
This increase in area was small, hence the creep curves

produced are assumed to be a constant load.

APPENDIX B

Data

109

TABLE 5
SEATING STRAIN OBSERVATIONS

 

 

 

 

 

Seating Strain Stress at Beginning of
Sample in/in J Elastic Response, psi
I-l .0105 75
I-2 .0079 105
I-4 .0018 76
I-5 .0057 96
I-6 .0055 92
I—9 .0050 80
I-10 .0050 90
I-11 .0064 106
I-l2 .0065 72
I-15 .0081 68
I-14 .0071 70
I-15 .0070 72
I-16 .0068 118
I-l7 .0104 195
I-18 .0075 129
I-19 .0101 165
I-20 .0096 180
I-Zl .0069 250
I-24 .0056 67
I-26 .0068 71
I-27 .0054 70
I-28 .0054 67
I-29 .0081 80
I-50 .0057 70

 

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