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values of V*, approximately three times the molar volume, and also large A S both shown in Table I seem to support this interpretation. In Table I, values of E, V*, and A S obtained from the present experiments are compared to those for diffusion in bulk ice single crystals. The value of V* for the diffusion coefficient of bulk ice was estimated by a conventional methodlo because no experimental data have yet been obtained. The growth rate of grains in polycrystalline ice obtained in this work is approximately of the same order as those obtained from deep ice cores in Antarctica.I2 However, it should be noted that the present experiments were only (12) P. Duval, M. F. Ashby, and I. Anderman, J.Phys. Chem., in this issue.
with homogeneous grain growth of which the driving force was limited to the boundary energy. The driving force in the depths of ice sheets must be complex with effects of stress due to the flow. In contrast to the enhancing effect of the residual stress, there is an impeding effect of air bubbles and inclusions. Anyhow, further careful comparative studies with deep ice cores are desirable. Acknowledgment. The authors are grateful to Dr. C. C. Langway (SUNY-AB) who operates the Central Core Storage Facility for providing us the Byrd Station ice cores. We thank to our colleague Mr. T. Hondoh for useful discussions. Financial aid for this work from the Ministry of Education, Science and Culture is acknowledged. Registry No. Water, 7732-18-5.
Creep of Ice as a Function of Hydrostatic Pressure Stephen J. Jones’ and Hemming A.
M. Chew
Snow and Ice Division, National Hydrology Research Institute, Environment Canada, Ottawa, Canada K1A OE7 (Received: August 23, 1982: I n Final Form: January 14, 1983)
Experiments have been performed on the creep of polycrystalline ice subject to a combination of uniaxial compression and hydrostatic pressure. The temperature was -9.6 f 0.15 “C, a dead weight load gave a uniaxial compressive stress of 0.47 MPa, and hydrostatic pressure was varied up to 60 MPa. The secondary creep rate measured was found to decrease slightly as the hydrostatic pressure was increased from 0 to 15 MPa, to pass through a minimum between 15 and 30 MPa and then to increase more rapidly from 30 to 60 MPa. This gave, therefore, an activation volume which varied from approximately +32 to -55 cm3 mol-’, indicating that more than one mechanism was controlling the creep rate.
Introduction The creep of ice has long been of interest for many reasons. It is normally tested very close to its melting point, allowing analogies to be drawn between it and high-temperature deformation of metals, or rocks in the earth’s mantle. The flow of water through temperate glaciers depends to some extent on the water in the intergranular veins’ and this, in turn, will depend on the hydrostatic pressure. It has always been assumed that the application of hydrostatic pressure increases the creep rate of ice. This assumption is based on two sets of experimental data,2v3 both somewhat limited. Rigsby deformed four single crystals in shear at 1and 300 bar hydrostatic pressure, and found that “the shear strain rate increased as the pressure was increased at constant temperature, but that the rate is practically independent of hydrostatic pressure when the difference between the ice temperature and the melting temperature is kept constant”. Haefeli and others reported results of three samples of polycrystalline ice deformed in compression at 0 and 29.5 MPa hydrostatic superimposed pressure. Again, “the creep rate is increased by the hydrostatic pressure at constant temperature, but reduced again below its original value if the temperature is dropped by the rate of the depression of the pressure melting point”. (1) J. F. Nye and F. C. Frank, “Symposium on the Hydrology of Glaciers”,I.A.H.S. Publ. No. 95, 1973, p 157. (2) G. P. Rigsby, J. Glaciol., 3, 271 (1958). (3) R. Haefeli, C. Jaccard, and M. DeQuervain, “I.U.G.G. General Assembly of Bern”, IAHS Publ. No. 79, 1968, p 341.
It is unfortunate that both these authors chose 0 and 30 MPa as their experimental pressures, because the natural assumption was then made that the effect would be linear between 0 and 30 MPa. In this paper we show that this is incorrect and that between 0 and about 15 MPa, hydrostatic pressure decreases the creep rate slightly and then, above 15 MPa, a minimum creep rate is reached followed by an increase in rate with increasing hydrostatic pressure. Met hod Samples of randomly oriented polycrystalline ice were made by admitting cold, air-free, water to a previously evacuated tube containing snow, and allowing it to freeze. The resultant sample sometimes had fine air bubbles running along its central core, but by cutting the samples lengthwise into four pieces, and then turning on a lathe, the central part was removed and the final sample, 10 mm in diameter by 30 mm long, was air-free and had a density of 0.917 mg m-3 (hO.001). The grain size of the sample was determined by observation of a thin section through crossed polaroids and was always close to 1.0 mm diameter. The samples had stainless steel end caps frozen on them and were then placed in a creep jig, which was put inside a high-pressure cell. Stress was calculated from a dead weight, compressive, load, and strain was measured with an LVDT, connected electrically through the high-pressure cell, and the hydrostatic pressure was applied with a mechanical hand pump, and measured with a Heise gauge. The equipment was inside a cold room maintained at a
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The Journal of Physical Chemistry, Vol. 87, No. 21, 1983 4065
Creep of Ice 30-
e
TABLE I: Approximate Values of Activation Volume, V, as a Function of Hydrostatic Pressure, P
U = 0.47 MPa T = -9.6 "C
V
e
e 7
20
-
P,MPa
P = 50 MPa
0-15 15-30 30-60
v)
c9
-
1 z d
-
e a
10-
._-ee-*
e -17
~O-~S-'
o
a
o oo 0
0 0 -
v
a
n
-
.
.
8.2
V = 0 47 MPa T=-96 C
0
o s-~
32
0.6
=0 - 55
- 1.0
b is the Burgers vector of (1150)dislocations in ice =
0.45 nm.
o o o
b3 a
cm3 mol-'
P
= 0.1 MPa
made. Again it can be seen that hydrostatic pressure, at first, decreases the creep rate, although it is not as pronounced as in the first type of experiment. The samples were examined after each test and contained no cracks.
Discussion The creep curves obtained were similar in shape to those previously reported for polycrystalline ice.4 First the strain rate decreased, primary creep, to a constant minimum value, secondary creep, before increasing again at strains of the order of 3%, tertiary creep. A strain of at least 1% , and sometimes closer to 2%, was always needed before secondary creep was established, and a minimum creep rate obtained. The values of minimum creep rate obtained at atmospheric pressure varied from 8 X lo4 to 11 X lo4 s-l. This variation is considered normal for such mechanical tests and probably reflects differences in initial mobile dislocation content from sample to sample. If we assume the minimum creep rate, imin, follows the rate equation kmin 0: exp[-(E + P V / k T ) ] (1) where E is the activation energy, P is the hydrostatic pressure, V is the activation volume, k is Boltzmann's constant, and T is absolute temperature, the activation volume is given by
It is related to the work needed to change the system in going from the "normal" to the "activated" state. Because it is usually very small, high pressure is needed to obtain accurate activation volume values. There is another expression in the literature for activation volume, namely (3)
where u is the applied stress rather than the hydrostatic pressure. This value, V, is easier to measure than V and many experiments have been carried out on different materials. However, in the present study, we measured V rather than V. It is clear from Figure 2 and eq 2, that V is not a constant but is a function of P, varying from a positive number for P I 15 MPa to an increasingly negative number for P 2 15 MPa. From the slope of the strain rate/pressure curve of Figure 2, the solid line, approximate values of V were obtained as shown in Table I, by reducing the curve to a straight line between the values of hydrostatic pressure indicated. The fact that V is not constant means that several processes are controlling the rate of deformation, rather than just one. At the high pressures, above 30 MPa, the (4) P.Barnes, D.Tabor, and J. C. F.Walker, Proc. R. SOC.London, Ser. A , 324, 127 (1971).
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temperature of the creep test is approaching the pressure melting point. A t these pressures, an increasing liquid phase at the grain boundaries, and grain boundary sliding, is probably responsible for the increasing creep rate.4 For small pressures, where V = +0.6b3, some other mechanism is responsible. This could be a dislocation climb mechanism, which typically gives V = l-10b3, or a defect reorientation me~hanism.~.~ However, the activation volume for defect reorientation, deduced from the dielectric relaxation time as a function of pressure,'t* is only =0.05b3 (cm3mol-') and so this mechanism would not seem to be ( 5 ) J. W. Glen, Phys. Kondens. Mater., 7,43 (1968). (6)R.W. Whitworth, Phil. Mug., A41, 521 (1980). (7) R. K.Chan, D. W. Davidson, and E. Whalley, J. Chem. Phys., 43, 2376 (1965). (8)R.Taubenberger,M.Hubmann, and H. Griinicher in 'Physics and Chemistry of Ice", E. Whalley, S. J. Jones, and L. W. Gold, Ed., Royal Society of Canada, Ottawa, 1973,p 194.
the one that is controlling the creep rate. The activation volume for climb, or for vacancy migration, has not been measured in ice. We hope to carry out further experiments a t different temperatures in order to understand the rate-controlling mechanism for creep in ice.
Conclusion Hydrostatic pressure less than 15 MPa, at -9.6 "C, cause polycrystalline ice to harden slightly and give an activation volume of N 32 cm3 mol-' (0.6b3). This is comparable to those found in other materials where the rate-controlling process is dislocation climb, but is ten times greater than the activation volume for defect reorientation in ice. Hydrostatic pressures greater than 15 MPa lead to a softening of the ice as the pressure melting point is approached and grain boundary effects become important. Registry No. Water, 7732-18-5.
Rate-Controlllng Processes in the Creep of Polycrystalline Ice P. Duval,*t M. F. Ashby,' and I. Andermant Laboratoire de GIaciologie et Gtkphysique de /'Environment, CNRS, B.P. 53 X, 38041 Grenoble CEDEX, France and Cambridge University, Engineering Department, Cambridge CB2 lPZ,England (Received: August 23, 1982; In Final Form: November 18, 1982)
The microscopic processes which control plasticity in polycrystalline ice a r e reviewed. Evidence is presented which indicates that basal slip does not control flow in polycrystals; it is the necessary deformation of other systems (of which four or five are necessary for polycrystal plasticity) which limits the rate of flow. The origins of kinematic and isotropic hardening, leading to anelasticity and transient behavior, are discussed. Recovery processes (permitting a steady state) and softening processes (leading to tertiary creep) are considered.
1. Introduction 1.I. Single Crystal and Polycrystalline Plasticity. A
single crystal of ice, loaded in simple tension or compression, will usually deform by slip (or creep) on the basal plane. Only by orienting it with great care is it possible to load the crystal in such a way that there is no shear stress on this plane; then its resistance to deformation is much greater. Data reviewed below show that nonbasal deformation of ice crystals requires a stress a t least 60 times larger than that for basal slip at the same strain rate. It is a matter of common experience that polycrystalline ice can be deformed to large strains. In both the flow of glaciers and the deformation of the antarctic ice cap, for example, plastic strains of more than 1 are usual. In laboratory tests in compression or torsion, strains of 20% are easily achieved. For this to be possible, each crystal in the polycrystalline body must have a t least four independent deformation systems available to it.'v2 Basal slip in ice provides only two independent systems. Polycrystal plasticity must involve slip or climb of dislocations on nonbasal systems; and because of its difficulty it must play a major role in controlling the macroscopic behavior. This macroscopic behavior is complicated by the existence of several competing mechanisms. All, ultimately, relate back to the properties of the individual crystals or of the boundaries between them. Before proceeding further, we review these mechanisms briefly. The symbols Laboratoire de Glaciologie et Geophysique de 1' Environment.
* Cambridge University.
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TABLE I : Symbols and Units principal stresses (MPa) principal strains principal strain rates (s-l) stress tensor (MPa) deviatoric part of stress tensor: sij = uij 6ijokk (MPa) strain (strain rate) tensor (s-l) equivalent stress (eq 1.1)(MPa) effective stress: u e = a - ui (MPa) internal stress (MPa) directional part of internal stress (MPa) nondirectional part of internal stress (MPa) equivalent strain rate (eq 1.2) (s-l) Young's modulus (MPa) time (s) temperature ( K ) grain or crystal size ( m ) molecular volume ( m 3 ) lattice diffusion coefficient ( m 2s - l ) boundary diffusion coefficient times boundary thickness (m3s - l ) power law creep constants ( s - ' , MPa, -) transient strain rate (s-l) fracture toughness (MPa ml'a) J K") Boltzmann's constant (1.38 x
used in the paper are defined in Table I. 1.2. Deformation Mechanisms i n Polycrystalline Ice. The macroscopic response of isotropic polycrystallinbe ice (1)J. W.Hutchinson, Proc. R. Soc. London, Sec. A, 348,101 (1976). (2)J. W.Hutchinson, Metall. Trans. A , 8,1465 (1977).
0 1983 American Chemical Society
