Publisher’s version / Version de l'éditeur:
Experimental Mechanics, 18, 12, pp. 464-470, 1978-12
READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright
Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.
Questions? Contact the NRC Publications Archive team at
PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.
NRC Publications Archive
Archives des publications du CNRC
This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. / La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur.
Access and use of this website and the material on it are subject to the Terms and Conditions set forth at
Rheology of columnar-grained ice
Sinha, N. K.
https://publications-cnrc.canada.ca/fra/droits
L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.
NRC Publications Record / Notice d'Archives des publications de CNRC:
https://nrc-publications.canada.ca/eng/view/object/?id=83f152f3-9f54-423b-95a1-8d1f7664ab4b
https://publications-cnrc.canada.ca/fra/voir/objet/?id=83f152f3-9f54-423b-95a1-8d1f7664ab4b
National Research
Conseil national
I
$
Council Canada
de recherches Canada
RHEOLOGY OF COLUMNAR*GRAINED ICE
by N.K.
Sinha
Reprinted from
Experimental Mechanics
Vol. 18, No. 12, December 1978
P*
464 *
470
DBR
Paper
No.
838
Division
of
Building Research
SOMMAIRE
L'article montre que le fluage de la glace 5 structure colomnaire
sous une force de compression nniaxiale perpendiculaire 2 la structure colomnaire comporte une rgaction glastique instantange suivie d'une d6formation retard6e Plastique et visqueuse. On indique aussi que ces dgformations Glastiques et visqueuses ont une Gnergie d'activation ggale. Cette glace peut donc 6tre prise comme matsriau "thermorh6ologiquement" simple dependant de la contrainte non lingaire. L'auteur a mis au point une proportion phenom6nologique simple qui peut 6tre employee pour des 6tudes approfondies sur la fonction de constante du fluage prgsent6e sous forme normalis6e.
Rheology
of
Columnar-grained Ice
Phenomenological viscoelasticity of columnar-grained ice has been experimentally investigated and analyzed in an effort to explain many apparent peculiarities of the
deformation behavior of polycrystalline ice
by N.K. Sinha
ABSTRACT-Creep of columnar-grained ice, under uniaxial compressive force normal to the columns, is shown to be composed of an instantaneous elastic response followed by a delayed elastic and viscous deformation. Both the delayed elastic and viscous strains are shown to have equal activation energies. Thus, this ice can be considered a s a thermo- rheologically simple material with a nonlinear stress dependence.
A simple phenomenological relationship has been developed that can be used for further analysis of the creep compliance function presented in a normalized form.
Introduction
Studies of the creep of polycrystalline ice have emphasized the steady state or secondary stage, giving little attention to the initial or transient creep range-the range of particular importance for many engineering problems involving ice. The reaction of structures t o a moving ice sheet, the safe use of ice covers under static and moving loads, and the use of ice-breakers for moving through ice- covered waters, are just a few of the problems confronting engineers that require a knowledge of the deformation and strength properties of ice. Recent activities in the colder regions of the earth demand a level of understanding of these properties that has not yet been attained.
The present report is concerned with the short-term rheology of columnar-grained ice under uniaxial com- pressive stress normal t o the columns. This type of ice and stress condition is common in many field situations. Although many types of ice are found in ice covers, it is considered that the results of this study will increase the engineer's appreciation of the behavior that can be expected in the field.
N.K. Stnha is Research Officer, Geolechnrcal Secrton, Dtvtston of Burlding Research, Nalional Research Council of Canada, Ottawa, Canada K I A OR6.
Paper was presenled at 1978 SESA Spring Meeltng, held tn Wichira, KS on May 14-19.
Original manuscript submitled : April 14, 1978. Ftnal version received :
July 3, 1978.
Preparation of Ice
Ice was made in a cold room at - 10°C from deaerated water using the technique described by Gold.' The water was allowed to cool in an insulated plastic-lined container ( I x 0.6 x 0.3 m) equipped with a pressure-relief system. Freezing was initiated by spreading finely crushed and aged ice on the water when its surface temperature was a few degrees above freezing. The ice obtained was about 20 cm thick and without any visible air bubbles.
Because of the method of initiating freezing, the ice crystals in the top layer were randomly oriented. Ice is a hexagonal crystal with a marked tendency t o grow more readily in directions perpendicular t o [0001] (c-axis) than parallel t o it. Because of this, grains with their axis of symmetry tending t o be perpendicular to the growth direction grew at the expense of less favorably oriented grains resulting in columnar-grained ice. The [0001] axis of these columns, however, tended t o have a random orientation in the plane parallel t o the-top surface, in other words, perpendicular t o the long directions o f the columns. This was structurally similar t o one of the more common types of ice formed from fresh water. Michel and Ramseier2 have designated this ice type as S-2. Natural-ice covers are subjected t o uniaxial freezing conditions. S-2 ice may be considered as transversely iso- tropic. The finite size of the ice-making container in the laboratory introduces a two-dimensional heat flow, except perhaps near the center. For this reason, the specimens were made from center ice.
The grain-size distribution and crystallographic orientation of individual grains, in a plane normal to the columns, can be determined by making thin sections and examining them with polarized light using a universal stage. The method, however, is tedious and time consuming.
A new etching and replicating technique3 was used to ascertain the transverse isotropy of the laboratory-grown ice and, thus, determined the choice of ice for specimens. Figure 1 shows an example of some grains and their corresponding [0001] axis. The average grain size of the specimens chosen was about 3 mm and the density was measured to be 0.915 ? 0.002 gm/cm3.
Preparation of Specimen
the transformers. A few repeated loadings, not exceedingRectangular specimens, 5 x 10 x 25 cm3, were
prepared so that the long direction of the grains was
perpendicular t o the 10
x
25-cm2 face. The specimenswere machined to their final dimensions by milling and stored in kerosene at - 1 0 ° C t o prevent sublimation. Tests were conducted a week or more after the preparation.
Experimental Procedures
Constant compressive loads were applied to the 5 x 10-
cm2 face with a simple lever-type loading system. The loading was thereby perpendicular to the long dimension of the columnar grains. The load was applied t o the specimen by lowering a n air-operated jack and was measured by a load cell. Time required t o apply the full load, without vibrating the system excessively, was about 0.2 s for loads less than 0.5 M N * m P . Time required t o apply full load was longer, however, for higher load levels.
Specimens were wrapped with a clear plastic sheet with some exposed areas for the gages, which were covered with vaseline. Creep was measured with a pair of specially designed strain gages containing linear differential transformers lightly attached to the 10
x
25-cm2 faces of the specimen with a spring mechanism. A special jig was used t o locate the devices on the specimen. Gage length was 15.25 cm, located at the central 3/5 of the block. The output of the transformers was recorded simultaneously on a high-speed strip-chart recorder and a digital data- logging system. Data-recording systems were kept outside the cold room.The differential transformers were calibrated at each 5 ° C increment in the temperature range - 5 t o -45 O C with an accurate displacement gage. Each transformer had a limited linear range of operation. Experiments were conducted so that maximum displacements were in this linear range. Strains could be measured with an accuracy of 2.5
x
Slight eccentricity in mounting the strain-measuring devices inhibited the free movement of the core inside
Fig. 1-Optical micrograph of replica of etched horizontal section of S-2 ice showing the random c-axis orientation of a few grains, as indicated by the arrows
more than 10 s, were conducted on each specimen, after mounting the displacement gages, to ensure that the transducer outputs did not vary more than _ + 5 percent from the mean output of the series for the same load at the same temperature. If variations were more than the allowed maximum, the positions of the transformers were adjusted to give more consistent readings.
~ x p e r i m e n t s were conducted in a cold room with temperature variation less than 0.25"C. The temperature of the room was changed in increments of 5 " C , and a period of 24 h or more was allowed for the specimen t o stabilize at the new temperature before any load application. A thermocouple in contact with the specimen was used t o
keep a continuous record of the temperature.
A series of tests was first conducted t o determine Young's modulus of the ice by loading and unloading t o several stress levels up to 2.0 M N m-2 in the temperature range - 5 to -45OC. The specimen was not kept under load, in any loading cycle, for more than 2 s.
Extensive observations were then made on creep under
a load of 4.9
x
lo5 N m-2 at various temperatures.Experiments were conducted first at - 4 5 ° C and repeated at the next higher temperature until a temperature of
- 10°C was reached. These tests were made on the same
specimen to avoid variations due t o the structure and arrangement o f the transducers. The load was left o n the specimen for about 100 s in each case. Both creep and creep recovery were recorded. At least two loadings were
applied at each temperature. An incubation period of 4 h
1
was allowed between loadings. After completing the i
foregoing measurements, loadings were repeated o n the same specimen with longer times of application. The experiments were terminated on a given specimen when
the accumulated nonrecoverable strain exceeded 2
x
t o avoid major irreversible changes in the structure of the ice.
A few representative creep and creep-recovery curves
for loads other than 0.49 MN m-' were then recorded at
various temperatures. The entire investigation is considered a long-term project and further developments will be described at a later date.
REDUCED TIME. s . AT 1 0 C
TIME, L . s
Result and Analysis
Young's Modulus and Initial Elastic Strain
Figure 2 shows creep and creep recovery at -4b°C for
a load of 0.49 MN m 2 , and a creep duration of 200 s.
The creep strain E: consisted of an instantaneous elastic
strain E: and a recoverable strain E : ; the superscript u
indicates the stress level. The creep strain was completely recoverable after load removal, within a time comparable to the time the specimen was under the load. Permanent deformation, if any, was less than the accuracy of measurement. Young's modulus can be determined from
where the subscript '0' corresponds to the initial strain,
E:, provided the load is applied instantaneously.
Figure 3 shows a typical set of readings at -40°C measured within 0.005 s after attaining the full load. Because there was always a finite time required to load the specimen fully, the evaluated modulus in Fig. 3 is
denoted as E,. The E, values obtained at temperatures
between -40 and -45°C were between 9.1 and 9.8
GN m-', with an average of 9.3 GN m-'. The scatter in the data was within the range of repeatability and error of measurement. These values are in the same range of
E, determined by high-frequency sonic for
polycrystalline fresh-water ice. High-frequency measure- ments of Yamaji and Kuroiwa7 in the temperature range
0 to
-
100°C and those of Gold6 in the range of -5 to-40°C indicate that Young's modulus of ice varies little (within a few percent) in the temperature range -5 to
-45 "C under consideration. Our values of E, determined at higher temperatures, were considered to be in error owing to the finite time of loading. This will be discussed later.
Delayed Elastic Strain
Recoverable strain shown in Fig. 2, was of elastic
origin but delayed in nature and, hence, the term 'delayed'
elastic strain. In this case, the delayed elastic strain E; was
about 40 percent of the instantaneous strain 6:.
Figure 4 gives three examples of creep and creep recovery at -30°C which also illustrates the repeatability
Fig. 2-Creep and recovery of S-2 ice at -41 'C, u = 0.49 M N W ~ - ~
of the observations. It was noted that breathing while examining the specimen closely after each load caused condensation on the core of the gages inhibiting their free
Fig. 3-Variation of initial strain with applied load at - 40" C for S-2 ice, compressed normal to the columns
"*=- 8
Fig. 4-Creep and recovery of S-2 ice at
-
-
4
-
30"
C for three successive loadings of-
0.49 MN-m-2 6; C VI
-
2 r 0 1001..
da--'+J--- 2 00 300 I 4001
5 0 0 TIME, I, 5movement. This was a chronic problem and could not be avoided completely in spite of taking all the precautions.
The observations showed that the delayed elastic strain was an exponential decay type of function with time. They also indicated that nonrecoverable deformation became significant if the load was applied for a sufficiently long time. The length of time required depended on the load and temperature. An example of measurable permanent deformation, E:, is shown in Fig. 5. In this
case, the recoverable part of the creep strain was almost equal to the initial elastic strain, compared to about 40 percent in Fig. 2.
Variation of the delayed elastic strain with temperature for a given stress was found to follow an Arrhenius type of relation, given by
where t , and t2 are the times required for a given strain at temperatures T, and T , , respectively; Q is the activation energy and R is the gas constant.
The series of experiments with the constant load of 0.49
MN m-2 applied at various temperatures was used to determine Q . The value of Q was found to be nearly equal to 67 KJ/mol (16 Kcal/mol). Experimental evidence
R E D U C E D T I M E . r . A T - 1 0 C
TIME. 1 , r
Fig. 5-Creep and recovery of S-2 ice at - 19.8 " C,
o = 0.49 MNom-2
of this observation will be clarified in the final analysis.
General Observation of Creep Strain
The observations suggested that the following general relationship would be an appropriate description of stress, temperature and time dependence of the creep behavior of columnar-grained S-2 ice :
where E : , e: and E; are as #defined earlier, and E: is the viscous deformation.
Viscous Creep Strain
The difference in creep strain at the moment of unloading and the total delayed elastic strain after complete recovery gives a measure of the viscous deformation during the time the load was applied. An irreversible viscous strain can then be determined, to estimate the average strain rate for this type of flow. Permanent strain, shown in Fig. 5, was 2.5
x
giving an average strain rate of3.1
x
s-I. This agrees well with the minimum creep rate reported by Mellor and Testa8 and Tabor and Walker9 for polycrystalline ice, and the steady-state rate reported by GoldLo for S-2 ice, when adjusted to the corresponding temperatures and stress levels. It should be pointed out, however, that this permanent deformation was near the accuracv of measurement of the mesent experiments. The example is cited mainly to demonstrate ;he probability that the viscous strain becomes measurable at approximately800 s at -20°C and that the corresponding creep rate can be reasonably estimated.
Temperature dependence of the steady-state creep rate f o r polycrystalline ice has been commonly represented, after Glen1', by
ig = A exp (- Q/RT) (4)
where A is a constant for a given stress. Goldlo used eq (4) to present his long-term creep data on S-2 ice for a load level of 9.8
x
lo4 N m-2. His observations were made on a single specimen over the temperature range of -5 to -40°C. He used the same type of ice under conditions of loading similar to those in the present experiments. Activation energy for steady-state flow was found to be 65 KJ/mol (15.5 Kcal/mol), which agrees well with the value for the delayed elastic strain o,btained from thisstudy.
Using eq ( 4 ) , the total viscous creep strain at time t after loading is given by
t : ( t ) = i;t = A t exp ( - Q / R T ) ( 5 )
If 1, and t2 are the times at TI and T , , respectively, for
the same amount of viscous strain, then a relation identical
i to eq (2) can be obtained from eq (5) for viscous flow.
Total Creep Strain
Equation (2) implies that, if the strain is plotted against the logarithm of time, then the time dependence of the
strain at temperature T , can be obtained by shifting the
measured dependence a t T, along the time scale by the
factor s, ,,. The ratio s,
.,
is often called the shift function.Since the activation energy for both the viscous flow and the delayed elastic deformation appears t o be equal, the same function can be used t o shift the viscous and the delayed elastic components of the creep curve in the strain-time-temperature space coordinates. This fact and the experimental evidence that Young's modulus or the initial elastic strain is relatively independent of tempera- ture, means that creep strains obtained at various temperatures can be combined to give a master curve at some standard temperature.
Creep curves for a single specimen of S-2 ice, for a few representative temperatures and a load of 0.49 MN m-',
shifted to a reference temperature of - 10°C, are shown
in Fig. 6 . It is considered that the scatter is due largely to errors associated with strain measurements.
The existence of the horizontal shift function implies that all relaxation or retardation times change by the same factor when the temperature changes. In the field of high polymers, this type of thermorheological behavior is known as the William-Landel-Ferry (WLF) Law.'? Any material that follows the W L F type of deformation behavior is known as a thermorheologically simple material. Figure 6 indicates that polycrystalline ice is in this category. It is to be noted that the portion of the composite-creep curve at the shorter end of the time scale can be obtained from data gathered a t low tempera- tures whereas the creep behavior for relatively longer
times can conveniently be obtained from data at elevated temperatures. Thus, the reduced curve can be obtained for a wide range of times in spite of the technical difficulties of measuring initial creep response at elevated temperatures, the temperatures usually associated with ice problems.
A more explicit form of eq (3) is
E: = E:
+
c (E:).' [ I - exp { - ( a T t ) b } ]+
&,"t (6)where b , c and s are constants, and a , is given by
where d is a constant. Equation (6) is a modified form of
a phenomenological relation used by Sinha" t o describe creep behavior of structurally stabilized plate glass at temperatures higher than the transformation range.
If a r - I and a r - ~ are the values corresponding t o T, and
T,
,
respectively, eq ( 7 ) givesComparison of eqs (2) and (8) gives
Thus the constant a , incorporates the shift function. Equation ( 6 ) can be presented in terms of stress level,
u, as
t: =
+
c (z)S [ I - exp { -+
i l tI
u (E. Eo
where i,! is given by eq (4) for unit stress, and
n
is thestress exponent.14
Equation (10) appears t o satisfy the experimental results. as shown by the solid line in Fig. 6 , with the following
values : 12 I
I
I I9-
10 --
Ln '0 8 --
M b: yl . 6 --
Z-
-&-%#+R~P%PO%*~*~ u DI - C A L C U L A T E D + e - 4 4 . 2 c L n 4-
0 - 4 1 C-
- 3 0 C 0 - 2 0 C + - 1 5 C 2-
r - 9 . 9 C-
0 - 9 . 9 C A - 9 . 9 " C 0I
I
I
I
II
1 0 - ~ - 1loo
Io1
l o 2 10) TIME. I. sFig. 6-Creep of S-2 ice reduced to a
reference temperature of - 10" C, a= 0.49
Q = 67 KJ/mol (16 Kcal/mol; 0.70 ev) c = 3 s = 1
n
= 3 b = 0.34 ( 2 l / n ) d =-
22.339 giving a2s30K = 2.5x
s-'t,! = 1.76
x lo-' s-' for 1.0 MN m-2 at 263 OK ( 1 1 )Discussion
It is evident from eq (10) that the initial shape of the creep curve is mainly determined by delayed elasticity. The intermediate range is governed by the relative contributions of both delayed elastic and viscous creep. With time, the delayed elastic component of the creep asymptotically approaches a limiting value. Long-term creep is therefore governed by the characteristics of viscous deformation. Equation (10) thus maintains consistency with previous long-term observations o n ice for conditions for which a steady state can be achieved involving n o significant structural changes, such as internal cracks and r e c r y s t a l l i ~ a t i o n . ~ The proposed model explains many observations that were thought t o be peculiar t o ice and which added confusion to the under- standing of the deformation behavior of this material. These will be discussed in the following sections.
Creep Rate
The variation of strain rate is given by differentiating eq (10) with respect to time,
cb a
::
= -(-)'
exp { -+
i,!I
u1
" (12)t E,
According to eq (12), with the valuesgiven in eq ( l l ) , the delayed elastic-strain rate is directly proportional to stress whereas the viscous creep rate is proportional t o the third power of stress. The former will therefore decrease with time more rapidly than the latter when the applied stress is increased. For example, the delayed elastic-strain rate, 5.6 h after application of a load of 0.1 M N m-2 at
- 10°C, is equal to the viscous-strain rate, but is only 4 percent of that strain rate for a load of 0.5 M N m-2 and 1 percent for one of 1.0 MN m-Z. Thus 'quasi'-steady states would be attained in the total creep-strain rate at shorter times for higher whereas a prolonged creep period may be required to reach the same state at lower stresses."'
D ( t ) = ~ ( t ) for u = 1 ( 1 3) The relationship between the creep-compliance function and the time-dependent modulus, E ( t ) , is given by
Ice is a nonlinear viscoelastic material and eqs (13) o r (14) in their present form cannot be used t o determine its behavior a t some other stress level, unless D and E are presented as functions of stress at a given temperature, i.e.,
This form is t o o complicated for a general analysis. A normalized form of the creep compliance function may be presented by
D." E: E,"
- = - - -
-
DP E," €p
where ' 0 ' indicates the response a t zero time, and E,"
represents Young's modulus, E,.
Computed normalized creep-compliance functions of S-2 ice for a few stress levels at - 10°C are presented in Fig. 7 . The dashed line is the compliance function calculated from the first two terms of eq ( l o ) , i.e., elastic response only. Comparison of the elastic response with the observed behavior at the given stresses shows that the function does not change appreciably with stress up t o 2 MN m-2 for time less than approximately 1 s. This time range is increased t o about 20 s when the stress is lower than 0.5 M N m-'. Variation of these times with temperature can be obtained by shifting the curves along the time scale by the shift function. Thus, for example, stresses up t o 0.5 MN m-2 would have little effect on the time-dependent modulus for time less than approximately 2000 s when the temperature is -45°C. Because these are also the time ranges when the contributions of the viscous flow to the total strains are negligible, the ice has essentially an elastic behavior in the sense that all the itpposed strains are recoverable.
Static and Dynamic Modulus
The variation in the measured values of the elastic modulus of ice determined by static and dynamic methods
Stress Exponent for Creep-strain Rate 1 . 0 ~ I I I I I
In the previous examples, the viscous creep rate was - -
assumed to have a stress exponent of 3. If stresses are C . Y -
low and experimental time insufficient t o reach the steady -
state, one might measure part of the transient-creep rate, -" in which case the measured stress exponent ,would be
bl-
'
'
- lower than the usual ~ a l u e . ~ ' ~ - ' ~ The experiqental time b 8 8 - -could very well be in the initial part of the transient "0.4
-
2.0 MPJlrn2...,,, 1.0 M N I ~ '
creep, in which case the creep rate will be governed by the ba" -
.- 0 . 5 M N l r n 2
delayed elasticity which has a stress exponent of unity. --- E L A S T I C RESPONSE Thus one could conclude that a Nabarro-Herring type of 0.2-
creep was being observed in ice a t small stresses. This -
aspect of ice creep has been discussed in greater detail.20
0 I I I I I I
l o 3 l o 0 10' l o 2 1 0
Time-dependent Modulus T I M E . I , s
For a linear viscoelastic material, the creep-compliance Fig. 7-Normalized creep-compliance function for S-2 ice function, D , can be uniquely defined as at - 10'C for different loads
I
0 I I I 1 I
1
,02 , 0 3 TIME, t. s
Fig. 8-Normalized creep-compliance function of S-2 ice at different temperatures for a load of 0.5 MN-m-'
can also be explained by eq (16) or its graphical presentation in Fig. 7. Figure 7 shows that measurements made in times less than s (5
x
102 Hz) a t ' - 10°C (or 0.1 s a t-
45 "C) will be very close t o E.. Experimental observations of Ewing el al,4 N o r t h w o ~ d , ~ Yamaji and Kuroiwa,' Gold6 and others support this and show little variation with frequency. As can be appreciated from Fig. 7, however, static methods will be subject t o uncertainties that depend o n time of measurement, temperature and stress. 16.''Consider strain and effective modulus at 5 s after application of full load for loads less than 0.5 MN m-'. The effective modulus will be lower than the high frequency or zero-time value (Fig. 8) by 6 percent at -45"C, 12 percent at -30°C, 23 percent at
-
10°C and about 30 percent near freezing point, numerically agreeing with Gold's measurements6 o n this type of ice. It will be lower by 10 percent at -45 "C, 20 percent at-
30°C, 35 percent at-
10°C and more than 50 percent at temperatures just below the freezing point, 1/2 min after application of the load. Similar errors will be involved if the time periods mentioned above are required t o apply the full load. The increase in apparent activation energy at temperatures near the freezing will also introduce discrepancies in the effective modulus in that temperature range.The first example (5 s for applying the load) simulates the condition of measurement readily realized in the laboratory and the second (load applied for 1/2 min) is not unusual for the field situation. Field values, therefore, will be generally lower than those determined in the laboratory. Both measurements will, however, show that the effective modulus increases with decreasing tempera- tures. A detailed analysis of this subject is given elsewhere.20
Summary
The initial rheological behavior of columnar-grained S-2 ice under uniaxial compressive force normal t o the columns, has been investigated. Time-dependent deforma- tion a t a given temperature is characterized by initial elastic response followed by delayed elastic and vjscous creep. Both delayed elastic and viscous strains are shown t o exhibit the same activation energy. Creep curves a t temperatures in the range - 10 to - 45 OC are shown to be normalized by a Williams-Landel-Ferry type of shift
function with a n activation energy of 67 KJ/mol (16 Kcal/mol). A simple phenomenological relationship has been developed t o describe the nonlinear stress dependence of this ice. The first two terms of the equation are shown to provide a model for the temperature and time dependence of the elastic (recoverable) modulus immediately after loading. An effort was made t o simplify the presentation of the rheological response in the form of a normalized creep-compliance function. Analysis indicates that the static effective modulus of ice will reflect the measuring conditions. It will decrease with increase of temperature and load. S-2 ice with average grain size of 3 mm can be considered a n elastic material, from the engineering point of view, for times less than 1 s at - 10°C for loads up t o 2 M N m-'.
Acknowledgments
The author is indebted t o D. Wright for his assistance in designing and conducting the experiments. This paper is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division.'
References
I. Gold, L. W., "The Process of Failure of Columnar-grained Ice," Phil. Mag., 26 (2), 311-328 (1972).
2. Michel, B. and Ramseier, R.O., "Classification of River and Lake Ice Based on its Genesis, Structure and Texture, " De'partement de Ge'nie
Civil, Universite' Laval, ~ u i b e c , Canada, Report S-I5 (1969).
3. Sinha, N.K., "Observation of Basal Dislocations in Ice by Etching and Replicating, " J. Glaciol., 21 (85), 385-396 (1978).
4. Ewing, M., Crary, A.P. and Thorne, A.M., "Propagation of Elastic Waves in Ice," Phys., 5 (6), 165-168 (1934).
5. Northwood, T.D., "Sonic Determination of the Elastic Properties of Ice, " Can. J. Res. Sect. A., 25 (2), 88-95 (1947).
6. Gold, L. W., "Some Observations on the Dependence of Strain on Stress for Ice," Can. J. Phys., 36 (lo), 1265-1275 (1958).
7. Yamaji, K. and Kuroiwa, D., "Viscoelastic Property of Ice in the Temperature Range 0 to - 100°C. " L o w . Temp. Sci., Ser. A,'I5, 171-183 (1956). (In Japanese, Defence Res. Bd. of Canada, Transl. T633, Ottawa, 1958).
8. Mellor, M. and Testa, R., "Creep of Ice under Low Stresses," J. Glaciol., 8 (52). 147-152 (1969).
9. Tabor, D. and Walker, J.C. F., "Creep and Friction of Ice," Nature, 228 (5267). 137-139 (1970).
10. Gold, L, W., "Activation Energy for Creep of Columnar-grained Ice, " Physics and Chemistry of Ice, E. Whalley, S. J. Jones and L. W.
Gold, eds., Roy. Soc. Canada, Ottawa, 362-364 (1973).
11. Glen, J. W., "The Creep of Polycrystalline Ice," Proc. Roy. Soc. London, A228, 519-538 (1955).
12. Williams, M.L., Landel, R.F. and Ferry, J.D., "The Temperalure Dependence of Relaxation Mechanisms in Amorphos Polymers and Other Glass Forming Liquids," J. Am. Chem. Soc.. 77 (14), 3701-3707 (1955).
13. Sinha, N.K., "On the Studies of Rheo-oplical Response of Plate Glass in a Wide Temperature Range," PhD Thesis, Univ. Waterloo, Waterloo, Canada (1971).
14. Langdon, T.G., "Creep Mechanisms in Ice," Physics and Chemktry of Ice, E. Whalley, S.J. Jones and L. W. Gold, eds., Roy. Soc. Canada, Orlawa, 356-361 (1973).
15. Barnes, P., Tabor, D. and Walker, J. C.F., "The Friction and Creep of Polycrystalline Ice," Proc. Roy. Soc. London, A324, 127-155 (1971).
16. Voitkovskii, K.E, "The Mechanical Propertk of Ice" (Mekhanicheski svoistva l'da, Moscow, Izd. Akademii Nauk SSSR, 1960). translated by Am. Meteorological Soc., Boston, MA, AFCRL-62-838 (1962).
17. Colbeck, S.C. and Evans, R.J., "A Flow Law for Temperate Glacier Ice, " J. Glaciol., 12 (64). 71-86 (1973).
18. Bromer, D.J. and Kingery, W.D., "Flow of Polycrystalline Ice a1 Low Slresses and Small Strains," J. Appl. Phys., 39 (3). 1688-1691 (1968). 19. Bufkovich, T. R. and Londauer, J. K., "Creep of Ice at Low Strmes. "
U.S. Army Snow, Ice and Permafrost Research Establishmenl (now CRREL), Res. Report 72 (1960).
20. Sinha, N.K., "Short-term Rheology of Polycryslalline Ice," J. Glaciol., 21 (85). 457474 (1978).
21. Hobbs, P. V., Ice Physics, Chrendon Press, Oxford (1974). 22. Glen. J. W., The Mechanics of Ice, Cold Regions Res. & Engrg. Lab.. Hanover, NH, Monograph 11-C2b (1975).
T h i s publication is being d i s t r i b u t e d b y the Division of Building R e s e a r c h of the National R e s e a r c h Council of Canada. I t should not b e r e p r o d u c e d i n whole o r i n p a r t without p e r m i s s i o n of the o r i g i n a l p u b l i s h e r . T h e Di- v i s i o n would b e glad t o b e of a s s i s t a n c e i n obtaining s u c h p e r m i s s i o n .
P u b l i c a t i o n s of the Division m a y b e obtained by m a i l - ing the a p p r o p r i a t e r e m i t t a n c e ( a Bank, E x p r e s s , o r P o s t Office Money O r d e r , o r a cheque, m a d e p a y a b l e t o the R e c e i v e r G e n e r a l of Canada, c r e d i t N R C ) t o the N a t i o n a l R e s e a r c h Council of Canada, Ottawa. K I A 0 R 6 . S t a m p s a r e n o t a c c e p t a b l e .
A l i s t of a l l publications of the Division i s a v a i l a b l e and m a y be obtained f r o m the P u b l i c a t i o n s S e c tion, Division of Building R e s e a r c h , National R e s e a r c h Council of Canada, Ottawa. KIA OR 6.
