Mechanical properties of polycrystalline ice: An assessment of current knowledge and priorities for research

CoM Regions Science and Technology, 3 (1980) 263-275 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

MECHANICAL PROPERTIES OF P O L Y C R Y S T A L L I N E K N O W L E D G E A N D P R I O R I T I E S FOR RESEARCH

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ICE: AN ASSESSMENT OF CURRENT

REPORT PREPARED FOR THE INTERNATIONAL COMMISSION ON SNOW AND ICE, WITH SUPPORT FROM THE U.S. NATIONAL SCIENCE FOUNDATION R. LeB. Hooke 1 and M. Mellor 2 (co-chairmen); W.F. Budd 3, J.W. Glen 4, A. HigashP, T.H. Jacka 6, S.J. Jones 7, R.C. Lile 3, R,T. Martin s, M.F. Meier 9, D.S. Russell-Head 1° and J. Weertman ~ (members of the working group) 1Department of Geology and Geophysics, University of Minnesota, Minneapolis, MN (U.S.A.) 2Cold Regions Research and Engineering Laboratory, Hanover, NH (U.S.A.) 3Meteorology Department, University of Melbourne (Australia) *Department of Physics, University of Birmingham, Birmingham (U.K.) SDepartment of Applied Physics, Hokkaido University, Sapporo (Japan) SAntarctic Division, Department of Science and the Environment, Melbourne (Australia) 7Glaciology Division, Department of Environment, OrPawa,Ontario (Canada) SDepartment of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA (U.S.A.) 9U.S. Geological Survey, Tacoma, WA (U.S.A.) ~°Faculty of Engineering, University of Melbourne (Australia) ~Department of Materials Science, Northwestern University, Evanston, IL (U.S.A.)

(Received February 29, 1980; accepted March 3, 1980)

PREAMBLE Systematic knowledge of the mechanical properties of ice is a fundamental requirement for the solution of a wide range of problems in the earth sciences and engineering. Past research has been characterized by specialized studies, in. volving a variety of motivations and scientific disciplines, without much overall coordination. When the results of this research are synthesized, a reasonably coherent picture emerges, but there are major gaps in knowledge. From the deduced behavior of ice, and from informed speculation where data are lacking, a coherent plan for future research can be developed. This report summarizes existing knowl. edge and proposes experiments and test programs for a new research thrust. The studies that are proposed can be expected to yield significant benefits in glacier studies, ice engineering, and the mechanics of frozen ground.

INTRODUCTION A workshop on the mechanical properties of ice was held in Boulder, Colorado, March 12-16, 1979; subsequent meetings with a broader group of specialists were held in Canberra and Melbourne, Australia, on December 10 and 15, 1979, respectively. The objectives were to establish the basic empirical facts regarding the mechanical properties of fresh-

water polycrystalline ice, and to focus attention upon critical areas for further research. Specifically excluded from detailed discussion were snow, sea ice, and frozen soils. The physical processes involved in creep and fracture were discussed in general terms where appropriate, but they were not considered in detail. Despite the large amount of work that has been done on the mechanical properties of fresh-water

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polycrystalline ice, our understanding of this complex material has many serious gaps. It is generally agreed that upon application of a constant stress, a sample of ice with a random fabric will show an initial elastic deformation followed by a stage of decelerating creep rate, and finally a stage of accelerating creep rate. The acceleration may be a result of: (1) the formation of microcracks; (2) recrystallization and the development of a non-random fabric; and/or (3) dislocation multiplication. For a given stress field and given strain, the creep rate will be a function of temperature, grain size, impurity content, and the nature of the stress field. It may also be necassary to take into account the previous history of the sample and the method of preparation. Additional research is needed to better define the shape of the strain-time curve as a function of stress, temperature, grain size, and impurity content; to establish the correspondence between strain-time curves in constant stress experiments and stressstrain curves in constant strain-rate experiments; and to determine the conditions under which the accelerating part of the strain-time curve may lead to steady-state creep rather than terminating in failure. These conclusions are discussed in greater detail in the body of the report, and more specific recommendations are given in the final section.

Terminology A few of the terms used in this report are defined here to avoid ambiguity, because they sometimes have been used in other ways in the past. emin is the minimum strain rate in "secondary creep" under constant stress. Omax is the upper yield stress, or strength, defined as the maximum stress reached during a constant strain rate test. o , or the residual strength, is the final strength in a constant strain rate test carried to large strains. ~o~ is the final value of the "tertiary" strain rate, the strain rate that would be achieved in a constant stress test carried to large strains.

CHARACTERISTICS OF ICE Only one type of ice (ice Ih) occurs widely in nature on earth, and that is the form discussed here. However, its mechanical properties differ

significantly depending on where and llow it was formed: ice-rich frozen soil is a different material. and behaves differently, from the relatively pure ice of glaciers, rivers, and lakes. The columnar crystals of lake ice may fail under stress in a different way than the relatively equi-angular crystals of snow-ice, or the complexly intergrown crystals of" glacier ice. Some polycrystalline ice is isotropic in its properties, but some consists of crystals that are highly oriented, so that it deforms several times faster in one direction than in another. The mechanical properties cannot be specified without specifying such other properties as impurity or inclusion content, texture, and fabric.

Impurities and inclusions Inclusions in ice can be gaseous (such as air dissolved in the ice lattice or as discrete bubbles), liquid (such as water occurring at 3- or 4-grain intersections), or solid (such as clay or sand particles). Above their eutectic temperatures, dissolved impurities are concentrated in the liquid phase. Such solutes may have relatively large effects, even in low concentrations, because they tend to accumulate at grain boundaries and thus influence grainboundary migration. Dissolved impurities can also occur as ions in the lattice. Because yielding under stress is often controlled by the nucleation and motion of dislocations, impurities or inclusions in the crystal lattice may serve as nucleation sites or as obstacles to the motion of dislocations, and so may influence the mechanical properties.

Texture The term texture refers to the sizes and shapes of the individual grains and inclusions in a polycrystalline aggregate. Natural ice crystals range in size from less than 1 mm to more than 100 mm, and in shape from approximately' equi-dimensional grains to long thin columns, fiat plates, or complexly intergrown shapes. Grain sizes and shapes, and the distribution of inclusions, can be relatively uniform, allowing a homogenous sample to be prepared, or they can be quite non-uniform and heterogenous. The texture of some types of ice, such as foliated glacier ice or lake ice, can be strongly anisotropic.

265 Fabric

The term fabric refers to the orientation of the crystallographic axes of individual crystals in a polycrystalline aggregate. A single ice crystal shears more than 10 times faster when the shear stress is applied perpendicular to the hexagonal axis than it does when the stress is parallel to this axis. Thus the degree to which a specimen yields under a given stress depends critically on whether the distribution of crystal axis orientations is statistically random, or whether a certain orientation is preferred, and in the latter case on how this preferred orientation is related to the orientation of the stress field of the testing environment. A preferred orientation (anisotropy) is produced naturally by many ice deposition or freezing processes. For instance, a fabric can be inherited from the recrystallization of a mass of oriented snow crystals. Subsequent deformation may be accompanied by recrystallization, either in nature or in a specimen under test, and tends to produce a fabric which reflects the geometry of the stress field. These deformation fabrics can be very strong, with an appreciable fraction of the crystal axes oriented very close to one, two, three, or four specific directions. Highly strained ice commonly exhibits this marked anisotropy.

(5) Heterogeneity of the sample in all respects (6) Material purity: (a) electrical conductivity of melt water; (b) percent inclusions and grain size distribution of solid inclusions (7) Storage conditions, including temperature and the time between sampling and testing (8) Preparation techniques, particularly for artificial ice (9) Past history of sample, including temperature and deformation history, particularly for natural ice.

STRESS/STRAIN/TIME RELATIONS Elastic deformation

The true elastic moduli of ice are difficult to determine from quasi-static experiments, but convincing values have been obtained by dynamic methods that subject the material to high frequency waves, or pulses, of small amplitude. Additional data would be welcome, especially data giving the effects of variations in temperature, gas content, dissolved impurities, solid inclusions, anisotropy, and the like. More data on compressibility under hydrostatic pressure would be useful, especially for compression to the phase boundary, and for compression at high rates for ice with inclusions or impurities.

EXPERIMENTAL MATERIALS

Creep deformation at constant stress

Experimental materials may be either naturally occurring ice or samples prepared in the laboratory. The size and shape of the sample to be tested depend somewhat upon the type of test. A minimum requirement is that the minimum linear dimension of the sample be 10 times the maximum grain size. In order to properly interpret stress-strain data for ice samples it is necessary to have an adequate description of the material. Descriptions for natural and laboratory samples should include, where appropriate, the following: (1) Geographic position of sampling site (2) Depth of sample, and temperature at that depth (3) Crystal fabrics, including orientation with respect to applied stress (4) Texture, not only of the ice but also of any inclusions

Time-dependent deformation by delayed elasticity and viscous flow is most frequently studied by making constant stress creep tests in uniaxial compression or tension. Data have usually been presented by plotting strain, e, against time, t (Fig. 1). In general, when plots of this kind are carried to sufficiently large strains (say >3%), they show: (1) an initial instantaneous elastic deformation; (2) a stage of decelerating creep (~ negative); (3) a stage in which strain rate appears to become constant for a time (~ -~ 0); and (4) a stage of accelerating creep (~ positive). The terms primary, secondary, and tertiary creep have been applied to stages 2 to 4, respectively. Secondary creep has, in the past, been referred to as "steady state creep", but in general secondary creep should more properly be regarded as a transition from the decelerating (strain hardening) stage

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1"-- 3 - - ~ J-- 2 " ~ / ~

/ . ~ . . . ~ s~opo Fa//ure strain (bt inflection poin/)

!

~t

Fig, 1. A typical strain-time curve for a constant stress test. Numbered stages are described in text.

to the accelerating (strain softening) stage. The secondary creep rate is the minimum strain rate (drain) for this test (Fig. 2), or the strain rate at the inflection point of the complete creep curve. Note that this inflection point gives the maximum value for the ratio of stress to strain rate. The accelerating processes leading to tertiary creep may be (1) crack formation; (2) recrystallization accompanied by the development of non-random fabrics; or (3) dislocation multiplication. In laboratory experiments on polycrystalline ice all three processes may occur. Process 3 occurs in laboratory studies on monocrystals. Process 2 is inferred to be particularly important in glaciers, and this is the basis for suggesting that steady-state tertiary creep may be the pertinent parameter in this case. The limiting creep rates for well-developed tertiary creep have not been well established, but there may be a limiting constant strain rate, ~ , if test geometry can be maintained (Figs. 1 and 2). This implies that

0"

2

"steady state" creep, if it exists at all, will be tile creep that develops at large strains when the ~ransie~lt deceleration and acceleration processes axe eithe~ exhausted or in balance. The distinction between dmin and d is emphasized in Fig. 2. In the past, there have been some serious errors in the determination of minimum strain rate, due largely to termination of tests at small strains (<<1%). Older procedures for data extrapolation, most notably application of the Andrade Law, now appear to be faulty. A useful recent development is presentation of creep data in the form of plots of log strain rate against either strain or log time. Such plots give a much clearer indication of minimum strain rate. Recent data suggest that the strain to emin is nearly constant (~1% axial) over a wide range of stresses and temperatures. This relation, if confirmed by further work, could be useful in designing future experiments. Additional studies are necessary at strain rates below 10 4 s-~, however. Some of the research needs in connection with this type of test are: (a) General awareness of the limitations inherent in small strain tests, and of the potential for error in data extrapolations; (b) Techniques for running tests to large strains (10% or so) without unduly distorting the test geometry ; (c) More data for creep to large strains, so as to establish whether tertiary creep tends to constant strain rate, and what changes in fabric and texture have occurred, and (d) Systematic collection of data on the strain at the inflection point of complete creep curves (which is perhaps best given by plots of log g against either e or log t). Deformation and yielding at constant strain rate

Cr"N

I ~Foilure

sfro/n

Fig. 2. This figure, derived from Fig, 1, emphasizes the distinction between ~min and ~ .

The most common test for determining the strength, or upper yield stress, of ice is the constant strain rate test in uniaxial compression or tension. Test results are usually given in the form of a stress/ strain (o-e) curve (Fig. 3). In general, the complete stress/strain curve from a careful test in a stiff (nonaccelerating) machine displays: (1) a nonqinear increase of stress with strain, the slope at the origin, (do/de)~=0, representing the true Young's modulus,

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hmit, and what changes in fabric and texture have occurred; and (c) Systematic collection of data for Omax and o®, and for the strain for Omax. Correspondence between results from constant stress tests and constant strain rate tests

_--E Fig. 3. A typical stress-strain curve for a c o n s t a n t strain rate test. N u m b e r e d stages are described in text.

or "initial tangent modulus"; (2) a peak where o is a maximum and do/de is zero; (3) a non-linear decrease of stress with strain, where d2o/de 2 changes sign and where do/de reaches a maximum and then gradually decreases in magnitude to a small value, or perhaps to zero, at some finite value of o. Curves of this form can be generated for both ductile yielding and brittle fracture, provided that the testing machine is not subject to uncontrolled acceleration when stored strain energy is released (that is, the machine must be either inherently stiff or controlled by a rapid response servo system). The peak of the stress/strain curve gives the maximum stress Omax that can be mobilized at the imposed strain rate, and this stress can be regarded as the peak strength of the ice, irrespective of whether the yielding is by flow or fracture. In other words, strength is defined by the maximum value of the ratio of stress to strain rate. The asymptotic limit of stress for large strains, if indeed there is a finite limit, could be regarded as the "residual strength", o . The spread between the peak strength, Omax, and the residual strength, o® (i.e. Omax - o**) decreases as the strain rate, ~, decreases, perhaps tending to zero at very low strain rates. Some of the research needs related to this test are: (a) Techniques for generating complete stress• strain curves to high strains (say up to 10% or so) for a wide range of strain rates, without unduly distorting the test geometry; (b) More data for deformation to high strains, to establish whether stress tends to an asymptotic

A number of investigators have assumed, without explicit justification, that there is a direct correspondence between the creep curve for constant stress and the stress/strain curve for constant strain rate. This seems to be a reasonable assumption, although there are some questions about the stress• strain/time histories in the two tests. For comparison of the two tests, consider the creep curve and the stress/strain curve in Figs. 1, 2 and 3. On the e-t curve, Young's modulus E is given by the applied stress and the intercept on the strain axis (although the usual plot on linear scales is unsuitable for defining this intercept). On the o-e curve, E is given by do/de at o=0, e=0 (provided that the scales of the plot are suitably chosen). These may not be very practical ways to measure E because of platen seating problems. Looking again at the e-t curve, the strain rate at the inflection point, emin, together with the applied stress o defines a maximum value of o1~. On the o-e curve, the peak stress Omax, together with the applied strain rate ~, also defines a maximum value of o/~. If (o/d)max is an intrinsic property of the material, then a plot of o against ~ ought to be the same curve for each of the two tests. If this is true, either test could be used to measure both minimum creep rate and peak strength, and possibly also the "failure strain" (Fig. 3). Such correspondence has not yet been established conclusively, either by theoretical models or by systematic experiments. Finally, a plausible correspondence can be deduced for the large strain limits of the two curves. On the e-t curve, ~ . gives a limiting value of o/~ for high strains. On the o-e curve, o** gives a limiting value of old for high strains. If ~** is finite on the e-t curve, then o should be finite on the o-e curve, and vice versa. If there is direct correspondence between the two tests, both should give the same value for (o/~) .

268 Another expectation would be that, if dmin ~6~ under very low constant stress, then Omax ~ o under very low constant strain rate. Such data as exist tend to support the correspondence that has been outlined, but clearly there are important questions to be settled in this matter. There is a need for systematic experiments that give curves of cr vs. gmin and of o vs. g for constant stress, and curves of Omax vs. i and of o vs. d for constant strain rate. Because the failure strain, or strain to 6rnin, appears to vary within quite narrow limits, the variation of failure strain with stress and strain rate needs to be established. It would also be desirable to have a mathematical model that completely describes both the e-t curve and the o-e curve when a common differential equation is solved for appropriate boundary conditions. Relations between stress and strain rate

The stress/strain-rate relation that has been of most interest to glaciologists is the relation between o and 6min for constant uniaxial stress, although in the light of the above discussion, the relation between a and the steady-state tertiary creep rate, d , is probably more important. The stress/strain-rate relationship that has been of most interest to engineers is the relation between Omax and d for constant strain rate. If the correspondence discussed above were to be. validated, then the first and third of these relationships would actually be the same, although the chief interests of glaciologists and engineers might lie at opposite ends of the strain-rate spectrum. The results of constant-stress creep tests at constant temperature can be represented by a simple power relation between 6min and or, 6min = A°n F o r the octahedral* stress range from 15 bars down

to 1 bar, and possibly as low as 0.2 bars, n -~ 3, which is in accord with theoretical expectations based on certain physical models. For these lower stresses, on the other hand, some experimental data suggest values of n lower than 3, with the decrease in n *roct =

~ [(o1_02)2 + (02_05) 2 + ( % - o l ) 2 ] 1/2 where ol, o 2,

and o a are the principal stresses.

occurring at lower stresses in tests at higher temperatures and in tests in simple shear. However, the low stress tests that gave values of n ~ 1 have now been discredited (they were short duration, small strain tests that did not decelerate to emin). There are still questions about the work that suggests that n ~ 2 for the low stress range. More data are needed for this range, but in order to establish emin unequivocally, the required tests must be of very long duration. A possible alternative, which still has to be completely validated, is to pre-strain test specimens at relatively high temperatures or stresses. At stresses above 15 bars or so, a logarithmic plot of 6min against o appears to curve, in the sense that n increases. When the results of constant strain rate tests at constant temperature are plotted on logarithmic scales, the Omax-4 curve for uniaxial compression at relatively high strain appears to merge smoothly with the o-drain curve for constant stress tests at lower stresses and strain rates, tending to support the correspondence argument. However, whereas the relationship for creep at low stresses and strains is insensitive to sign (that is, it is much the same for tension and compression), the curve splits into separate branches for uniaxial tension and compression at high strain rates, say above 10 -6 s-1, at temperatures within 10°C of the melting point. For strain rates higher than about 10 -s s -I, tensile strength, OT, does not change much as b increases; that is n ~ oo. The compressive strength, o c, continues to increase as d increases, so the ratio Oc/OT increases from unity towards values comparable to those expected from fracture theory, e.g. Oc/OT = 8 in simple Griffith theory. The available data relating Omax to g at high strain rates (above 10 -3 s -l) still leave some uncertainties, with some recent work contradicting earlier indications of a drop in Omax above the ductile-brittle transition. There are no data for either Oc or OT at the very high strain rates developed in impact loading, but such data, perhaps generated by Hopkinson bar or gas gun tests, would be useful for setting upper limits to the overall o-~ relations. From theoretical considerations, it is conceivable that Crc could continue to increase with g up to a limit of the order of 2 kilobars. Although past interest has centered on the o-g

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relationship that represents the maximum ratio of o [ ~ , future work should be designed to give a o-~ relationship that represents the high strain values of o and ~. This relationship would probably be the one to give steady-state creep conditions and long term residual strength. Some of the research needs in this area are: (a) Convincing experiments to establish the stress dependence of emin at very low constant stresses; (b) Systematic tests, both constant stress and constant strain rate, that go to large strains so as to give data on ~o and o o ; (c) High quality strength tests at high strain rates, using equipment that is capable of tracking complete stress/strain curves; and (d) Special strength tests at impact loading rates.

Among the problems related to such ice that merit investigation, the following are perhaps of primary importance: (a) The variation of the pressure melting point with the concentration of chemical impurities needs to be established. (b) Techniques are needed to measure the amount of the liquid phase present. (c) Techniques are needed for carrying out experiments at temperatures very near the pressure melting point. A particular problem associated with such tests is that of the variation in stress within a sample, and hence of the variation in the pressure melting point within the sample. (d) The variation in emin with impurity concentration and with concentration of the liquid phase needs to be established.

T E M P E R A T E ICE TEMPERATURE

The term "temperate ice" is often used to refer to ice at the pressure melting point*. A more precise, though still arbitrary boundary between temperate and cold ice is that temperature at which half of any heat removed from the ice is latent heat released during freezing of a liquid phase, and half is sensible heat released by cooling of the (pure) ice. Owing to the presence of impurities, phase changes begin to occur at temperatures slightly below the pressure melting point of pure ice. Ambiguity still exists, however, because the properties of ice at temperatures as low as -10°C suggest that there is a gradational transition from cold to "temperate" ice. Despite this ambiguity, ice near the melting point is important in many practical studies, and some specific investigations designed to elucidate its properties are needed. Some limited experimental data suggest that strain rate triples with an increase in water content from 0 to about 0.8%.

*The term "pressure melting point" is used loosely herein. In a rigorous analysis, the changes in pressure in the liquid phase and in the solid phase must be considered separately, as they are not necessarily equal and the change in the melting point is, in part, a function of the difference between them.

Effect on strain rate

It is generally agreed that at temperatures below -10°C, the effect of temperature on strain rate can be described by an Arrhenius relation, = ~o e - Q / n r

(1)

where d is the strain rate at absolute temperature T (K), eo is a reference strain rate (literally the strain rate at T = ~), Q is the activation energy for creep, and R is the gas constant. Above -10°C this relation predicts strain rates that are too low by as much as a factor of 5, presumably due to the increased importance of grain boundary slip as the amount of melting along grain boundaries increases, but possibly because eo and Q are functions of temperature and stress, respectively, as can be seen by comparing eqn. (1) with the complete equation for a rate-activated process. In using eqn. (1) it is tacitly assumed that eo and Q are nearly constant over the range of temperature and stress that are of interest, but this may not be the case. The use of an Arrhenius relation for laboratory work at temperatures below -10°C is reasonable, as this relation is based on sound physical principles. However, use of eqn. (1) as an empirical representation of the temperature dependence of

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strain rate at temperatures above --10°C should be discontinued, and more accurate empirical equations representing the curvature in plots of In ~ vs. 1/T should be sought. In view of the common assumption that grain boundary effects are responsible for the curvature in In g vs. 1/T plots, it is reasonable to expect that the degree of curvature might be a function of grain size and chemical purity of the ice. Investigations of these effects are needed, preferably as part of a more thorough study of the exact cause of this non-linearity. If temperature is not held strictly constant during a test, the strain-rate (or stress) will fluctuate, and it is not clear that the resulting mean value will be that appropriate to the mean temperature. This phenomenon should be investigated in its own right, but its presence emphasizes the need for accurate temperature control. In particular, a watch should be kept for oscillations with the frequency of the cooling cycle when analyzing records. Effect on strength

The strength of ice at a given strain rate (o at (a/4)max , Figs. 1 and 3) increases with decreasing temperature, a fact that is obvious from the aforementioned decrease in minimum strain rate with decreasing temperature. However, because of the nature of the relationship between Omax and d, the temperature dependence of Ornax is much weaker than the temperature dependence of 4min at constant stress. High quality data are sparse for high strain rates, and more experimental work is needed. The rate of increase, do/dT, decreases with decreasing temperature, but do/dT may not decrease uniformly. Some data suggest a discontinuity in the o-T relation at about - 3 5 ° to -40°C, with a small but abrupt increase in strength there. Further data are needed to confirm whether this is a real effect.

GRAIN-SIZE EFFECTS Effect on creep rates

There is little systematic information on grain size effects in the literature. Above a grain size of

1 mm several workers have found that the creep rate increases approximately with the square of the grain size. Many of these tests were not continued to train, however; preliminary results of recent tests that reached fmin suggest little or no influence of grain size on ~min. Only one investigator has reported measurements on ice with grain sizes less than 1 ram, and he obtained a surprising relationship in which the secondary creep rate drops by a factor of 5 as d increases from 0.5 mm to 1.0 mm (that is, g c~ d~-3). This was attributed to Nabarro-Herring creep, but at the high stresses (~5 bar) and strain-rates (~1 yr q) used, it seems unlikely that was the rate controlling process. Further systematic study of this problem, preferably over a greater range of grain sizes, is needed to clarify these puzzling and in some cases contradictory results. Standardization of procedures for determining grain size is desirable. The average diameter of cells in a thin section underestimates mean grain size and is not relevant for comparison with theoretical models, but it is an easy parameter to measure. For a better estimate of actual grain diameters in samples with equant grains, maximum chord-intercept methods may be used. Effect on strength

A systematic investigation of the effect of grain size on the strength of ice is needed. Theoretical studies and some limited experimental data suggest that tensile strength decreases as grain size increases, as is the general trend in the behavior of other brittle solids.

INCLUSIONS AND IMPURITIES

Certain chemical impurities are known to have a large effect on the flow of single crystals of ice. They would also affect, perhaps to a lesser degree, the flow of polycrystalline ice, but no systematic studies of this effect have been made. Such impurities are probably of greatest importance at temperatures above the eutectic temperature of the mixture. However, regardless of the test temperature, it is

271 recommended that when making samples in the laboratory, water which has been distilled and then deionized, rather then tap water, be used to avoid any possible effect of the ionic impurities. Inclusions of solid particles decrease ~min and presumably increase Omax. Some data suggest that this arises, in part, from a reduction in grain size of the ice with increasing inclusion content. The inclusions are concentrated on grain boundaries and it is inferred that they thus inhibit grain growth. Smaller inclusions may be more effective than larger ones in this regard, but we are not aware of any systematic studies of the effect of inclusion size. Gas bubble inclusions, on the other hand, increase the strain rate and lower the strength. Thus, it appears that the compliance and viscosity of the inclusion may also affect strain rate and strength. A dividing line between dirty ice and frozen soil may be taken as the ice concentration at which the solid inclusions cease to be surrounded by ice. More data are needed to cover the entire range in volume fraction of inclusions and to investigate the effect of inclusion size.

EFFECTS OF ANISOTROPY

We consider here two types of anisotropy, those due to fabric, and those due to texture, including the orientation and distribution of impurities or inclusions such as bubbles or dirt particles. Creep rate

Textural anisotropy due to the distribution of impurities, as in foliated ice, is best considered as a problem in the mechanics of the separate types of ice, taken individually, and will not be considered further here. Textural anisotropy due to the orientation of inclusions probably does not have a significant effect on creep rate unless the concentration of oriented inclusions is higher than commonly found in natural ice. We are not aware of any experimental data bearing on this problem. Textural anisotropy due to ice crystal shape may have a significant effect on creep rate in situations in which grain boundary slip is important, as would be the case at high temperatures. We are not aware

of any experimental data confirming such an effect, however. Fabric anisotropy has a marked effect on creep rate, but some ambiguity exists in rigorously describing this effect because fabric anisotropy is often a product of deformation. As discussed above, it seems that the increase in strain rate from emin to steady-state tertiary creep, ~**, may be a result of recrystallization and the development of a preferred orientation of c-axes. Thus in a constant stress test on polycrystalline ice with an initially random fabric, the first fairly constant strain rate obtained would be emin, whereas in a test on a sample with a strong preferred orientation it could well be ~0.. The ratio between the two strain rates derived from such a pair of tests has been called the enhancement factor. Some limited laboratory tests indicate that the enhancement factor is dependent on the relative symmetries of the fabric and the stress field. These tests gave enhancement factors of 1.4 for ice with a small-circle fabric subjected to unaxial compression, and up to 4 for ice with a strong single maximum fabric in simple shear. A theory has been developed which seems to predict enhancement factors reasonably accurately. A complication arises if a sample with a nonrandom fabric is subjected to a stress field with a symmetry other than that of the fabric. (For example, a sample with a strong single-maximum fabric developed in simple shear might be subjected to uniaxial compression.) In this case, ~min (and by correspondence Omax) would be different from that obtained from a test on a sample with a random fabric. A further complication arises if a sample with a non-random fabric, say from a polar ice sheet, is subjected to a stress field which is intended to reproduce the symmetry of the strain in the glacier itself. If a sample with a strong single maximum fabric is thus tested in simple shear, for example, and the sample is misaligned by 11 degrees from the "easy glide" orientation, it has been calculated that the resulting octahedral shear stress could be enhanced by a factor of about 1.8 with respect to that calculated on the basis of the applied shear stress. Non-random fabrics develop in ice during flow. Common fabrics developed in cold ice range from small circle patterns in uniaxial compression to patterns with one, two, three, or four maxima in

272 simple shear. In general the symmetry of the fabric reflects the symmetry of the stress field. It has been suggested that the multiple-maximum geometry is a result of twinning. The independent variables controlling this fabric development appear to be stress, temperature, and cumulative strain. If this is the case, it may be possible to determine the stability fields for various fabrics in stress-temperature-cumulative strain space for a given stress field and ice density. Alternatively fabric may also be dependent upon the sequence of changes in temperature and stress. Associated with these changes in fabric are changes in texture, with coarse interlocking crystals being characteristic of multiple maximum fabrics, and finer more equant crystals characteristic of weakly-oriented or single-maximum fabrics. Techniques are needed for characterizing the size and shape of the coarse interlocking crystals. The processes involved in this recrystallization and grain growth are poorly understood. Of considerable importance is the question of memory; for example will two otherwise identical tests, one starting with a fine-grained randomly-oriented sample and one with a coarse-grained sample with a multiple maximum fabric, both give the same value of coo and the same fabric if carried to sufficiently large strains, particularly if the stable fabric under the test conditions is different from both of the initial fabrics? Or alternatively, to what extent would naturally occuring fabrics be reproduced by experiments conducted at c o n s t a n t stress if such experiments could be carried to very large strains? Recrystallization not only occurs during a test but also after the test is completed and the stress (or strain-rate) has been removed. The grain size and fabric after such recrystallization may well be different from those developed under test, and this will affect subsequent tests on the same specimen. More work is needed to investigate this phenomenon and also the possibility of recrystallization if the stress (or strain-rate) are markedly reduced rather than removed altogether. Throughout this discussion of fabric only the orientation of the crystallographic c axes has been considered. The effect of a-axis orientation is not expected to be of comparable importance but must not be totally neglected. In particular, the properties

of ice with a very strong single pole of c-axis orientation might be thought to be essentially those of J single crystal unless a-axis differences are taken into account. More work is needed to examine the difference in behavior between such ice and monocrystatline ice. In view of the importance of fabric anisotropy in ice deformation problems, several recommendations seem in order: (a) In presenting fabric data the original uncontoured fabric diagrams should be used. (b) The percentage of c-axes falling in one percent of the area of an equal area net is not a statistically significant parameter and use of this measure of fabric strength should be abandoned in favor of methods based on sounder statistical principles. (c) Further tests of the enhancement-factor theory are desirable, using a greater variety of natural fabrics. The configuration of the applied stress field should be documented in such tests. (d) Additional effort is needed to clarify the conditions under which various textures and fabrics are stable, or the sequence of conditions that produce the various fabrics. (e) The recrystallization that occurs on reduction or removal of the load needs more thorough investigation. (f) The effect of grain boundaries on ice deformation needs more detailed study, with particular emphasis on ice in which the c-axes are essentially parallel but the a-axes are not.

Strength Textural anisotropy due to the distribution of inclusions may have some effect on the strength of ice, but we are not aware of any experimental data bearing on this question. Textural anisotropy due to ice crystal shape has an important effect on residual strength and fracture, particularly at temperatures above -10°C where melting along grain boundaries becomes important, and particularly in multiaxial stress fields. This is a subject receiving increasing attention by engineers concerned with floating ice having a columnar structure, and it is of considerable practical importance. The effect of textural anisotropy on ductile yield stress is less well known. Some weakening is to be

273 expected at relatively high temperatures, as just noted, but we are not aware of any data on such effects. The effect of fabric anisotropy on strength is ambiguous because, if the correspondence principle is correct, the maximum stress in a constant strain rate test would be Omax for polycrystalline ice with a random fabric, but could be the residual strength for ice with a strong preferred orientation of c-axes.

M U L T I A X I A L STRESS

It is generally assumed that under a given stress the components of strain-rate are proportional to the stress deviator, and that the second invariant of the strain-rate tensor is a function only of the second invariant of the stress deviator tensor. This assumption may not be universally applicable and laboratory experiments will have to be done to determine under what circumstances it is valid. In particular, strain rate may well depend upon the first invariant, in addition to the second invariant, especially at high stresses. The third invariant may also be a factor governing the strain rate of anisotropic ice. Effect of the first invariant

The effect of a confining pressure on the creep rate of ice has been investigated briefly; at strain rates of about 10 -a s-1 and temperatures near -10°C, a hydrostatic pressure of 400 bars increases the creep rate of polycrystalline ice by 20%. The effect may be larger for single crystals. When temperatures are measured relative to the pressure melting point, the confining pressure has no effect, or a slight hardening effect, depending on the data considered. Clearly a more systematic study of this effect is needed. The effect of hydrostatic pressure on the strength of ice at high strain-rates (~10 "4 s-1) is better documented, and shows that a confining pressure of 300 bars will approximately double the compressive strength of polycrystalline ice. At greater confining pressures the strength begins a gradual decrease. Effect of the third invariant

The third invariant of the stress tensor is a measure of the stress configuration.

There have been no systematic studies of the effect of the third invariant on either creep rate or strength. Some early work using combined torsion and compression has been interpreted as indicating that the constant of proportionality between a component of the stress deviator tensor and the corresponding component of the strain rate tensor is not a function of the second invariant alone, but more recent work supports the contrary view. Because most constitutive equations relating stress to strain rate assume that only the second invariant needs to be considered, further study of this effect is desirable. Biaxial stress fields

Biaxial stress fields in which one principal stress is zero are a type of multiaxial stress field which is of great practical importance for engineering problems involving floating ice. Detailed discussion of this problem is beyond the scope of the present workshop but is to be considered by a separate working group. Yield criteria for flow and fracture

Up to now there has been relatively little concern over formal yield criteria for the flow and fracture of ice, but such criteria are important for analysis of problems involving multiaxial stress states. For creep at low stresses it has long been assumed, with certain qualifications, that hydrostatic pressure has no effect on the relationship between the second invariants of stress and minimum strain rate. This implies a yield criterion similar to the von Mises or Tresca criteria, with the yield surface in three-dimensional principal-stress space disposed symmetrically about the hydrostatic pressure line and at a constant distance from it. However, the qualification is that allowance must be made for depression of the melting point as hydrostatic pressure increases. If constant temperature is stipulated, then the yield surface has to be a cone whose apex meets the hydrostatic line at the phase transition pressure, provided that the temperature is within the appropriate range for this type of behavior. For rapid creep and for brittle fracture at high strain rates, it appears that moderate hydrostatic pressure gives a corresponding increase in shear

274 resistance. For this range of behavior, the failure surface could be described by a non-linear MohrCoulomb criterion. However, limited data indicate that the "stiffening" effect of hydrostatic pressure eventually reaches a limit, and thereafter additional increases of hydrostatic pressure cause the failure surface to taper back towards the hydrostatic line. The main research need in this area is more triaxial compression data, covering a very wide range of strain rates.

experimental results show an opposite trend. Recen! studies support the latter view. The available KIC data can be considered to be only of a preliminary nature. There is no systematic study of variation of KIC with grain size, texture. fabric, impurity content, inclusion content, or morphology. The conflicting results on the tenlperature dependence of Kic need to be clarified.

SUMMARY AND RECOMMENDATIONS FRACTURE TOUGHNESS OF ICE (KIc)

A physical parameter that determines the condition for catastrophic crack propagation in a solid (under plane strain conditions) is the term KIC. A crack in a solid will propagate catastrophically when:

o(Traf/2f>~ KIC

(2)

where o is the applied tensile stress, a is the half length of the crack, and f is a calculable specimen geometry correction factor of order one. (For a crack in an infinite solid f = 1 .) For a perfectly brittle solid:

This report reviews the existing knowledge on the mechanical properties of ice and attempts to synthesize data from experiments conducted under a variety of experimental conditions. Gaps in the state of knowledge are identified and recommendations for further work are made. Specific recommendations have been mentioned throughout the body of the report. Here we reemphasize the most important of these. The recommendations fall into two categories: those in part A are for improvement of experimental technique and data presentation, and those in part B are for additional experimental work. Part A

l-V-

Where v is a Poisson's ratio, E is Young's modulus and 7 is the true surface energy of the solid. Substituting into eqn. (3), the smallest plausible value of KIC for ice is found to be approximately 0.45 MN m -aa. Available experimental measurements of Ktc for ice range from 0.05, the perfectly brittle case, to >0.3 MN m -3a. Values of KIC larger than that of a perfectly brittle solid have their origin, presumably, in the plastic deformation that can occur at the tip of a crack. Thus, under physical conditions in which plastic deformation is made more difficult the KIC value should be smaller. For ice it is reasonable to expect that if either the temperature is lowered or the strain rate is increased (or both) the value of KIC will lie closer to that of a perfectly brittle solid. Increasing the strain rate causes KIC to decrease in value. However, some experimental results indicate that reducing the temperature increases the value of KIC but other

(1) The secondary or minimum creep rate is, in general, not a steady-state creep rate and should not be referred to as such. (2) The use of the Arrhenius equation as an empirical description of the temperature dependence of the creep rate at high temperatures could be misleading. (3) The use of Andrade's law to estimate (~min from tests with low total strain is highly questionable. (4) Fabric data should be presented using the original scatter diagrams rather than contoured diagrams. (5) Plotting experimental results in terms of log vs. log t or log d vs. e serves to focus attention on the reliability of data and has merit. Part B

(1) Convincing experiments are needed to establish

275 the stress dependence of Cmin at stresses between 0.1 and 1.0 bars. (2) The correspondence of constant strain rate and constant stress experiments needs to be firmly established. (3) Techniques are needed for running tests to large strains (10% or so) without unduly distorting test geometry in order to establish, among other things, whether tertiary creep actually does tend to a constant strain rate, and stress to an asymptotic limit. (4) High quality strength tests are needed at high strain rates, using equipment that is capable of tracking complete stress-strain curves. (5) Further studies of the effect of temperature on strain rate at high temperatures are needed to investigate the effects of grain size, chemical purity, and rate controlling mechanisms. (6) Further systematic studies of the effect of temperature on dmin and d= are needed. (7) The stability fields of various fabrics, or the sequence of conditions necessary for their forma-

tion, need to be better defined, and the relevant independent variables identified with greater assurance. (8) The effects of the first and third invariants of the stress tensor need to be investigated more thoroughly. (9) Further tests of the enhancement factor theory are desirable, using a greater variety of natural fabrics.

ACKNOWLEDGEMENTS

Financial support for the workshop was provided by the U.S. National Science Foundation (Grant DPP 79-02795) and by the International Commission on Snow and Ice of the International Association of Hydrological Sciences. Facilities for the workshop were provided by the Institute of Arctic and Alpine Research, University of Colorado, Boulder, Colorado, the Antarctic Division, Department of Science and the Environment, Melbourne, Australia, and the University of Melbourne, Australia.