COLD REGIONS SCIENCE AND ENGINEERING Monograph ll-C2b

THE MECHANICS OF ICE

John W. Glen

December 1975

GB 2401 CORPS OF ENGINEERS, U.S. ARMY .U58m COLD REGIONS RESEARCH AND ENGINEERING LABORATORY no.ll-C2b HANOVER, NEW HAMPSHIRE 1975

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The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized docume* s. 0 Unclassified BUREAU OF RECLAMATION pENVER UBRARY C/ 92099625

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Monograph II-C2b -> 5. TYPE OF REPORT ft PERIOD COVERED 4. TITLE (end Subtitle)

j m MECHANICS OF ICE 6. PERFORMING ORG. REPORT NUMBER

8. CONTRACT OR GRANT NUMBER*» 7. AUTHORf» European Research Office r J.W. Glen ,r Contract DAJA37-68-C*0208 £ ■ '« • ' 1 "• - TO. PROGRAM ELEMENT, PROJECT, TASK 9. PERFORMING ORGANIZATION NAME AND ADDRESS AREA ft WORK UNIT NJUMBERS Dr. John W. Glen Department of Physics ^ DA Project 1T062112A130 University of Birmingham ( ^ Task 01 ____ Birmingham. England ____; '____ ; ______— 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Office, Chief of Engineers ^ December 1975 u 13. n u m b er o f p a g e s ' Washington, D.C. 47 14. MONITORING AGENCY NAME ft ADDRESS*?/ different from Controlling Office) 15. SECURITY CLASS, (of thie report) UvS ^Army Cold Regions Research and Engineering Laboratory-^ Unclassified Hanover, New Hampshire 03755 15«. DECLASSIFICATION/ DOWNGRADING SCHEDULE

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19. KEY WORDS (Continue on reverae aide if neceaaary and identity by btock number) Crystals Loads (forces) Engineering Mechanics Failure Growth (general) Ice ______-______■■ ■ ■ " ------— ------20. ABSTRACT (Continue on reverae atdd If neceawery and identify by block number) This monograph summarizes knowledge of the mechanics of ice. It is concerned principally with the, effect of stress on the mechanical properties of ice, including elasticity, anelasticity, sound propagation, plastic deformation and creep in single crystals and in poly crystalline ice, fracture, and recrystallization and grain growth that accompanies plastic deformation. The monograph also includes a comprehensive bibliography.

DD 1 jSS*78 U73 ew Tio w O F I NOV e* is o b s o l e t e ______Unclassified______SECURITY CLASSIFICATION OF THIS PAGE Data Entered) n

PREFACE

This monograph was prepared for the U.S. Army Cold Regions Research and Engineer­ ing Laboratory (USA CRREL) by Dr. John W. Glen, under a contract issued and adminis­ tered by the European Research Office, U.S. Army. The author is Reader in Ice Physics in the Department of Physics, University of Birmingham, England. Valuable suggestions and comments on the manuscript were made by Dr. Kazuhiko Itagaki and Dr. Malcolm Mellor, of USA CRREL, who served as technical reviewers. The monograph was written under DA Project 1T062112A130, Cold Regions Research, Task 01, Applied Research and Engineering.

The manuscript was completed in June 1970 and reflects knowledge prevalent at that time. References to a few later papers were included during subsequent revision. HI

CONTENTS Page Abstract...... i Preface...... -...... ii Editor’s foreword...... v Introduction ...... • ...... 1 Chapter I . Elasticity and propagation of sound in ice...... «...... « 1 Elasticity...... *.....------1 Propagation of sound in ice...... •...... 4 Chapter II. Anelasticity and damping of mechanical vibrations in ice...... 6 Chapter HI. Plastic deformation and creep of ice — single crystals ...... 10 Creep of ice single crystals...... 11 • Constant strain-rate tests...... 13 Plastic deformation of ice crystals in non-basal glide...... 14 Effect of surface conditions on plastic flow of ice crystals ...... 14 Effect of an electric field on plastic flow of ice crystals...... 15 Effect of impurities on the flow of ice single crystals...... 15 Slip lines and slip bands in ice...... 17 Effect of stress perpendicular to the slip plane on basal slip of ice crystals .... 18 Stress relaxation in ice crystals...... 18 Hardness of ice crystals...... 19 Chapter IV. Plastic deformation and creep of polycrystalline ice------.... 19 Results of creep tests on poly crystalline ice ...... •• 20 Shape of the creep curve...... 20 The flow law variation of strain rate with stress...... • 22 Variation of creep rate with temperature...... 24 Constant strain-rate tests...... 26 Stress relaxation...... — ..... 26 Strain relaxation...,...... 27 Effect of different stressing systems...... 27 Physical processes occurring during the creep of polycrystalline ice...... 28 Chapter V. Fracture...... 29 Tensile tests...... 30 Ring tensile and bending tests...... 31 Compression tests...... 31 Shear and torsion tests...... 31 Other tests on ice fracture...... •• 32 General summary of fracture behavior...... 33 Chapter VI. Recrystallization and grain growth of ice...... 33 Recrystallization...... *— ...... 33 Grain growth in ice...... 35 Literature cited...... 36 Appendix: Crystallographic indices used to specify planes and directions in ice 43 ILLUSTRATIONS Figure ^aBe 1. Matrix of coefficients of the elastic compliance for a hexagonal material such as ice Ih in axes in which the c-axis of the crystal is chozen as z-axis...... 2 2 The effective Young’s modulus as a function of stress orientation...... 5 3. Relation between relaxation time of pure H20 and D20 ice and the reciprocal of absolute temperature...... 7 4. Temperature dependence of internal friction in pure polycrystalline ice 9 5. Temperature dependence of internal friction in NaCl-doped ice crystals...... 10 6. Creep curves of ice single crystals deformed under various stresses at -70°C ...... 12 7. Stress-strain curves of ice single crystals obtained at a strain rate of 2.7x 10‘7 s"*, and at various temperatures ...... 13 8. Stress-strain curves for non-basal glide of ice single crystàls with various orienta­ tions of the 0-axis...... 14 9. Creep curves and stress-strain curves of ice single crystals at — 70°C with various concentrations of HF, compared with pure specimens...... 16 10. A bent ice crystal viewed by shadow photography showing the slip bands...... 17 11. Low-stress creep curves for fine-grained ice whose crystal orientation was initially random...... 21 12. Creep curves of randomly oriented polycrystalline ice at —0.02°C under various stresses at various temperatures under 6 bars ...... 22 13. Axial strain-rate as a function of axial compressive stress for fine-grained, ran­ domly oriented ice at — 2.06°C ...... 24 14. Logarithm of strain-rate as a function of 1 ¡T for poly crystalline ice, also for single crystals compressed perpendicular to their c-axes...... 25 15. Limiting envelope giving maximum value of stress/strain-rate ratio for uniaxial stress on isotropic polycrystalline ic e ...... 32 V

EDITOR S FOREWORD Cold Regions Science and Engineering consists of a series of monographs written by specialists to summarize existing knowledge and provide selected references on the cold regions, defined here as those areas of the earth where operational difficulties due to freezing temperatures may occur. Sections of the work are being published as they become ready, not necessarily in numerical order but fitting into the following plan, which may be amended as the work proceeds« The mono­ graph series was planned and directed by F.J. Sanger as editor until 1970, and has been directed thereafter by T.C. Johnson.

I. Environment A. General — Characteristics of the cold regions 1. Selected aspects of geology and physiography of the cold regions 2. Permafrost (Perennially frozen ground) 3. Climatology .a. Climatology of the cold regions: Introduction, Northern Hemisphere I b. Climatology of the cold regions: Northern Hemisphere II c. Climatology of the cold regions: Southern Hemisphere d. Radioactive fallout in northern regions 4. Vegetation a. Patterns of vegetation in cold regions b. Regional descriptions of vegetation in cold regions c. Utilization of vegetation in cold regions B. Regional 1. The Antarctic ice sheet 2. The Greenland ice sheet

II. Physical Science A. Geophysics 1. Heat exchange at the ground surface 2. Exploration geophysics in cold regions a. Seismic exploration in cold regions b. Electrical, magnetic and gravimetric exploration in cold regions B. Physics and mechanics of snow as a material C. Physics and mechanics of ice 1. Snow and ice on the earth’s surface 2. Ice as a material a. The physics of ice b. The mechanics of ice 3. The mechanical properties of sea ice 4. Mechanics of a floating ice sheet D. Physics and mechanics of frozen ground 1. The freezing process and mechanics of frozen ground 2. The physics of water and ice in soil vi III. Engineering A. Snow engineering 1. Engineering properties of snow 2. Construction a. Methods of building on permanent snowfields b. Investigation and exploitation of snowfield sites c. Foundation and subsurface structures in snow d. Utilities on permanent snowfields c. Snow roads and runways 3. Technology a. Explosions and snow b. Snow removal and ice control c. Blowing snow d. Avalanches 4. Oversnow transport B. Ice engineering 1. River-ice engineering a. Winter regime of rivers and lakes b. Ice pressure on structures 2. Drilling and excavation in ice 3. Roads and runways on ice C. Frozen ground engineering 1. Site exploration and excavation in frozen ground ? 2. Buildings on frozen ground 3. Roads, railroads and airfields in cold regions 4. Foundations of structures in cold regions 5. Sanitary engineering a. Water supply in cold regions b. Sewerage, and sewage disposal in cold regions c. Management of solid wastes in cold regions 6. Artificial ground-freezing for construction D. General 1. Cold-weather construction 2. Materials at low temperatures 3. Icings developed from surface water and groundwater

IV. Remote sensing A. Systems of remote sensing B. Techniques of image analysis in remote sensing C. Application of remote sensing to cold regions THE MECHANICS OF ICE

by

John W. Glen

INTRODUCTION

This Monograph is a companion volume to another Monograph in the same series dealing with the physics of ice. The purpose of the Monograph is to provide an introduction to the present un­ derstanding of the mechanics of ice for engineers and scientists who may need to work with or use it. For this reason the various chapters are as far as possible self-contained, and presume different levels of prior knowledge. The aim is to provide information about the mechanical properties of ice on the assumption that the general physical properties being discussed are understood by the reader, but that he has no knowledge of the way ice behaves. A reasonably detailed bibliography of recent papers is included. Older work is not discussed in the same detail, partly because some of it is unreliable but also because it is cited in the more recent work. This Monograph is concerned with ice, and since there are in the same series other Monographs dealing with the physics of ice,36 the physics and mechanics of snow,1 85 and with the mechanical properties of sea ice,126 those subjects will not be discussed here. For the most part this Monograph will be concerned with the mechanical properties of ice, that is to say elasticity, plasticity, fracture, propagation of sound, and recrystallization and grain growth that accompany or follow plastic deformation.

CHAPTER I. ELASTICITY AND PROPAGATION OF SOUND IN ICE

Elasticity Ice Ih being a hexagonal structure, its elastic properties are represented by a tensor with five in­ dependent components.96 That is to say if a stress a# is applied to the crystal, the resulting strain will be Cÿ, where the components of the strain tensor are related to the components of the stress tensor by the equations

e// “ Sijklakb (0

Here the convention that repeated suffixes are summed over has been adopted, i.e. eq 1 represents a series of equations, each of which has nine terms on the right-hand side, one for each of the pos­ sible combinations of/= 1,2 or 3 and / = 1,2 or 3. Conventionally96 this proper tensor equation is “contracted” into an equation connecting single-suffix stress and strain components joj and €/, where how the i run from 1 to 6, and where the relation between an applied stress and the resulting strain is expressed by the equation 2 THE MECHANICS OF ICE

*11 *12 *13 0 0 0

0 *12 *11 *13 0 0

*13 *13 *33 0 0 0

0 0 0 *44 0 0

0 0 0 0 *44 0

0 0 0 0 0 2 ( * n

Figure 1. Matrix o f coefficients o f the elastic com­ pliance for a hexagonal material such as ice Ih in axes in which the c-axis o f the crystal is chosen as z-axis.

€i = SijOf (2) where the summation convention now means that the right-hand side consists of six terms for each of the six components of stress. The reason why there were nine terms in the first equation and only six in the second is that the stress components o 1 2 and a 21 are equal to one another and have been combined into a single term a6, and the same applies for the other terms for which i does not equal /. One result of this is that the are not necessarily equal to the corresponding s ^ j, and in fact the full set of relations is that s^m = smn when m and n are 1 ,2 or 3; 2sijkl = smn when either m or n is 4, 5 or 6; and 4stjki = smn when both m and n are 4, 5 or 6. A result of the crystal symmetry of ice Ih is that many of the s#*/ or equivalently the stj are inter­ related, or, in crystal axes, zero. If the crystallographic c-axis is taken as the z-axis of the coordinate system, then the can be represented in matrix form (Fig. 1), and it can be seen that there are, as stated above, five independent components, i.e. sn , si2, si3, s33 and s44. These can be thought of as follows. Sn gives the extension perpendicular to the c-axis due to a longitudinal (tensile) stress perpendicular to the c-axis in the same direction. s12 gives the extension perpendicular to the c-axis due to a longitudinal stress also perpendicular to the c-axis but also perpendicular to the direction the extension is measured in. Sj3 gives the extension perpendicular to the c-axis due to a longitudi­ nal stress along the c-axis, and it also gives the extension along the c-axis due to a tensile stress per­ pendicular to the c-axis. s33 gives the extension along the c-axis due to a tensile stress along the c- axis, and S44 gives the shear strain in a plane containing the c-axis due to a shear stress in the same plane. It will be noticed that the remaining case, a shear strain in the plane perpendicular to the c- axis due to a shear stress in the same plane, is governed by a term which is twice the difference be­ tween two terms we have already had. This relation is entirely similar to the relation between Young’s modulus, shear modulus and Poisson’s ratio for isotropic solids, a relation proved in most elementary books on properties of matter. It is important to note that the matrix only takes this form in axes determined by the orienta­ tion of the crystal. In particular, if a longitudinal stress is applied in a direction neither perpendicular THE MECHANICS OF ICE 3

< l ,, -i -,i’ : - I ■ ' nor parallel to the c-axis, the results are much more complicated, and can best be deduced by rotat­ ing the tensor components (i.e. the sijkl with four suffixes) using the standard methods of crystal physics.96 Equations 1 and 2 are the correct equations to use if the situation is one in which a stress is ap­ plied and the resulting strain is being sought. If on the other hand it is required to know what stresses produce a given strain, it is better to work with the reciprocal relation

°ij = c ijkl €kl or, in contracted notation,

Oj = CijEj. ( 4 )

In this case no factors of two enter into the relations between r#*/ and the corresponding be­ cause a factor of two enters into the definition of ey in terms of eki when / is greater than 3 (i.e. when k ? 1). The matrix for the c/y is exactly similar to that for the s,y except that instead of 2(sn - sn) in the bottom right-hand corner, we have 1/2(cu - cn ). The values of these constants have been determined for ice at various temperatures 0 in °C using an ultrasonic pulse method.20 The results are as follows:

cn = 1.2904(1 ~ 1.489x 1(T30 - 1.85x l(T602)x 101* N n fa±0.3% c,2 = 0.6487(1 - 2.072x 1O'30 - 3-62x lO'602)x 1° 10 N m*2±2% Cu = 0.5622(1 - 1.874x lO'30)x 1010 N m‘2±7%

C33 = 1.4075(1 - 1.629xlO"30 - 2.93x 10"6fl2)x 1010 Nm’2±0.4% C44 =0.2819(1 - 1.6OlxlO‘30 — 3.62x lO'60a)x 1010 N m'2±0.7% From the above, the values of s(/- can be deduced:20

su = 1.040(1 + 1.070x 1O"30 1 1.87x lO'603)x 10*10 m2 N' 1 ±1% sn = -0.442(1 + 0.463x 1O'30 - 2.06x lO'602)x 10‘ 10 m2 N*1 ±6% Sl3 =,_0.189(1 + 1.209x 1O*30 + 6.15xlO*602)x 10'10 m2 N‘*±20% s33 = 0.848(1 + 1.405x 1O"30 + 4.66xlO"602)x 10*10 m2 N'1 ±1% S44 = 3.342(1 + 1.505x 1O'30 + 4.04x lO'602)x 10' 10 m2 N_1 ±1% From these values the compressibility can also be calculated; it is

k = 1.194(1 + 1.653xlO*30 + 3.12xlO"602)xlO'10 m2 N'*±15%

It will be noted that the value of Cu is much less accurate than the other c^, and that consequently those Sjj which involve c%3 are also less accurate. Another earlier determination of the elastic con­ stants of ice using a different technique (the Schafer-Bergmann technique which measures the opti­ cal diffraction patterns of ice crystals vibrating at high frequencies) gave values of the elastic para­ meters at -16°C which are reasonably consistent with those above.44 These values are:

cu = (1.3845 ± 0.08)x 1010 N m'2 c, 1 « (0.707 ± 0.12)x 1010 N n f2 4 THE MECHANICS OF ICE

C l 3 = (0.581 ±0.16)xl010 N m“2

C33 = (1.499 ±0.08)xl0'° N m '2

C44 = (0.319 ±0.03)xl010 N m '2

*11 = (1.04 ± 0.03)x 10"10 m2 N '1

S12 = <-(0.43 ± 0.03)x 10*10 m2 N '1

*13 = -(0.24 ± 0.01)x 10'10 m2 N*1

*33 = (0.85 ± 0.04)x 10'10 m2 N"1

*44 = (3.14 ± 0.03)x 10"10 m2 NT1

K = (1.11 ±0.07)xl0*'0 m2 N '1

The above values are, of course, those appropriate for ice in single-crystal form. Poly crystalline ice will have elastic properties that are some average of these, depending on the relative amounts of different crystal orientations in the poly crystalline aggregate. Ice which has all its c-axes aligned, such as certain samples of lake ice, should behave in very much the same way as a single crystal, and the above values would be appropriate, but for any other poly crystalline ice different values will be found, and these would have to be determined either experimentally or by making the appropriate average over the single crystal values. The extent of possible variation may be judged by noting that the value obtained for Young’s modulus in a typical experiment in which a tensile stress is ap­ plied and the corresponding strain parallel to the stress is measured is the reciprocal of sn parallel to the basal plane, and the reciprocal of s33 perpendicular to the basal plane, while for intermediate positions where the tensile axis makes an angle 0 with the c-axis it is the reciprocal of

Sit = S33 cos40 + snsin40 + ($44 + 2si3 )cos2 0sin2 0. (5)

This is plotted as a function of orientation in Figure 2, and it will be seen that the value varies by about 30%, and that the minimum value is smaller than either that parallel or perpendicular to the c-axis.

Propagation of sound in ice The propagation of sound in ice is of course governed by its elastic behavior. In general, in a single crystal there are three acoustic velocities for any one wave normal, corresponding to three polarizations of the sound wave, one quasi-longitudinal and two quasi-transverse. Only in particular crystallographic directions are these polarizations strictly longitudinal and transverse, but they are always mutually at right angles to each other. For details on the kinds of phenomena which occur in anisotropic crystals in general, a book such as that of Hearmon50 can usefully be consulted. When the crystal is oriented so that the wave normal is along a crystallographic direction of high symmetry, then the polarizations are longitudinal and transverse. In particular, with the wave nor­ mal in the c-axis direction, the longitudinal wave has a velocity determined by c33 and the trans­ verse waves are degenerate (i.e. all transverse polarizations have the same velocity) with the velocity determined by C44. Since elastic properties of hexagonal crystals are symmetrical about the c-axis, for any wave-normal perpendicular to the c-axis the polarizations are again strictly longitudinal and transverse. In this case the longitudinal wave velocity is determined by cn , the transverse wave polarized with its displacements parallel to the c-axis has a velocity determined by C44, and the transverse wave with its displacements perpendicular to the c-axis has a velocity determined by c THE MECHANICS OF ICE 5

c-oxis

Figure 2. The effective Young’s modulus E' = 7/s'n as a function o f stress orientation. Units are 109 N r n 2 = 104 bar. (After ref. 26.)

These are the only directions for which the waves are strictly longitudinal and transverse, and it will be noted that measurements of their velocities give directly all the elastic constants except ci3, since C66 = 1/2(cn - cl2). It is for this reason that the ultrasonic method gives these constants with greater accuracy than c13. In fact, a determination of the velocity of the quasi-longitudinal wave in some other direction, such as having the wavefront in the (1121) plane (see Appendix), gives a value for c 13, but the expression for it is more complicated and the value obtained less accurate. Again the behavior of polycrystalline ice will depend on the preferred orientation of the crystals of which it is composed. Ice with all the c-axes aligned should behave very much like the single crys­ tals, but other fabrics require some averaging procedure.108 In each case above where the sound velocity is said to be determined by a particular elastic con­ stant, the relation is that the wave velocity is yjcjp where c is the elastic stiffness coefficient con­ cerned and p the density of the ice. The variation of the velocity of sound with pressure up to 500 kbar has been studied for colum­ nar ice;8 most of the effects are attributed to recrystallization of the ice specimen. The elastic properties at low temperatures have been considered by Proctor,99 whose measure­ ments are in some respects apparently more accurate than those reported above. However his meas­ urements of shear wave velocities were restricted to the temperature range 60-110 K by experimen­ tal difficulties, and so are probably not so reliable for elastic constants in the higher temperature range. However his values down to 60 K are probably the best for extrapolation to lower tempera­ tures, and the values he obtains at absolute zero are:

cn = 1.710 ± 0.006x 1010 N m~2 6 THE MECHANICS OF ICE

c n = 0.851 ± O.OIOx 1010 N m '2 C13 = 0.713 ± 0.004x 1010 N m"2

c33 = 1.821 ± 0.001 x 1010 N m“2

C44 = 0.362 ± 0.001 x 1010 N m’2

From these he deduces a value for the average velocity of transverse sound waves of 2.128x 103 m s“1, and for longitudinal waves of 4.205x 103 m s’1. Finally these values can be used to deduce a Debye temperature of 223.6 K, a value which agrees very well with Debye temperatures deduced from X-ray Debye-Waller factors and specific heat measurements.

An anomaly in the elastic constants at about 105 K has been reported.51 This may be associated with the ordering of hydrogen atoms in the ice structure (see ref. 36, Chapter I). At very high frequencies the velocity of acoustic waves ceases to be constant and a dispersion is found. A study of this dispersion gives considerable information on the lattice dynamics of the crys­ tal, so studies of this dispersion are often made. Ice is however quite a complex case for several rea­ sons. The fact that there are many atoms in the crystallographic unit cell means that vibrations in which the different atoms or molecules move in opposite directions have to be considered. Such motions give rise to the so-called optical branches of the dispersion curves. Now ice has four water molecules in the crystallographic unit cell,.and«o has noTewer than 12 branches in its dispersion curve, even without considering motion of the hydrogen atoms independently of the oxygen atoms. When we come to consider this further development, yet another complication enters because of the random nature of the position of the hydrogen atoms (see ref. 36, Chapter I). Because of all this complication, it is not surprising that many of the theoretical studies have used simplified models - two-dimensional equivalent structures, cubic ice, etc. Experimentally, the curves can be deduced from neutron scattering,* and also from Raman and infrared studies discussed in Chapter VIII of ref. 36. A preliminary report on the theoretical work concerning the vibration of disordered systems of the ice type 21 111 indicates that the disorder has a very marked effect on the frequency spectra. The way in which phonons (i.e. quantized sound waves) interact with the protons has also been considered theoretically25 in a one-dimensional model. The low-frequency end of the frequency spectrum has also been deduced theoretically from a simple model of the ice structure with two force constants;24 the results compare quite well with the spectrum deduced from neutron scattering.

CHAPTER II. ANELASTICITY AND DAMPING OF MECHANICAL VIBRATIONS IN ICE

The response of ice to an applied stress is not completely elastic, that is to say there is not simply a strain which is completely determined by the stress and which disappears when the stress is re­ moved. If stresses are applied slowly, the phenomenon of plastic deformation occurs, in which a strain is produced which remains even after the stress has been removed; the same applies if larger stresses are applied for a shorter time to some forms of ice. This phenomenon will be discussed in Chapters III and IV. If too large a stress is applied relatively suddenly, the ice may fracture. This is discussed in Chapter V. But even if the stress is quite small and applied over a very short period,

•Refs. 2,47,48, 81, 93,98, 103, 116. THE MECHANICS OF ICE 7

Reciprocal of Absolute Temperature (K) 1

Figure 3. Relation between relaxation time o f pure H2 O and D2 O ice and the reciprocal o f absolute tem­ perature. (After ref 80.) as occurs when an acoustic wave is propagated through ice, the strain may not be completely in phase with the stress. This phenomenon, in which there is no permanent deformation, but there is some loss of energy, is known as anelasticity. In ice single crystals the anelastic phenomenon, which is responsible for the damping of sound waves, is pafticularly simple. The logarithmic decrement 8' for mechanical vibrations of the sample varies with frequency co at a given temperature according to the relation

2cor S9= - 8 ’ 1 + c o V

This is the form for a simple relaxation phenomenon in which the relaxation time is r and a maxi*, mum loss occurs when co = r"1. The value of r, and hence of the frequency for which damping is a maximum, depends strongly on temperature, while the value of 8fmax does not, although it varies considerably according to the crystal orientation and kind of acoustic wave.74 79 80 109 The varia­ tion of r with absolute temperature T follows an Arrhenius equation

r = roexp(0AT) where k is Boltzmann’s constant, but the values of activation energy Q found by the earlier experi­ ments74 79 109 were not consistent. This has subsequently been shown80 to be due to impurities in the ice; for pure ice80 109 the activation energy Q is 0.57 eV (Fig. 3), within experimental error the same as that found for dielectric relaxation, which also has the form of a simple relaxation phenom­ enon (see ref. 36, Chapter V). It therefore seems quite likely that this anelasticity in ice is due to the same process as is the dielectric relaxation, that is to say to the movement of L-defects. The theory of this process has been worked out by Bass6 7 and this theory is summarized in English by 8 THE MECHANICS OF ICE

Fletcher.26 Basically the idea is that, although in unstressed ice Ih the possible positions for hydro­ gen atoms (protons) are all equally favored (see Chapter I of ref. 36), when the ice is subject to a stress the positions are no longer all equivalent, and there will be energy to be gained by ordering the protons, just as when there is an electric field present there is energy to be gained by aligning the water molecules parallel to the field. The two kinds of arrangement to which the ice adjusts - are of course different, but in both the mechanical and the electrical case the final state of least free energy is determined thermodynamically by the applied stress or field and by the temperature, and the water molecules will turn to attain this state if they can. The role of the L-defects is that they migrate through the crystal, and as they do the water molecules rotate. At very low frequencies, therefore, the ice is always able to keep in the thermodynamic equilibrium state; at high frequencies the circumstances change too rapidly for the ice to be able to make any adjustment at all; but at in­ termediate frequencies, approximately equal to the reciprocal of the relaxation time, the water mole­ cules do undergo rotation, but not fast enough to keep up with the applied stress, and an energy loss results. In the electric case the polarization produced by the rotation is large compared with that which occurs due to other causes, and so the phenomenon is obvious from a study of the magnitude of the permittivity. In the mechanical case, the strain due to the proton rearrangement is small com­ pared with the strain of the crystal lattice as a whole, so no large change in elastic constant is observed. The theory predicts precisely what is found, that the relaxation time will be equal to that for di­ electric relaxation, that the maximum damping will be unaffected by temperature or impurity con­ tent, that those impurities which affect dielectric properties will also affect anelasticity, and that transverse waves will be much more strongly damped than longitudinal waves. In fact the theory predicts that longitudinal waves with wave-normal parallel to the c-axis should not be damped at all; experimentally the damping is an order of magnitude less.

A study has also been made of the anelastic relaxation of D20 ice.80 133 The activation energy Is practically the same as that of H20 ice (0.575 eV) but the value of r 0 is about double (1.04x 10" 15 s compared with 6.9x 10‘ 16 s) so that the frequency of maximum damping for any given temperature is about halved. Figure 3 shows the results for relaxation time on both H2 O and D2 0 ice compared with some dielectric results. This figure can also be used to deduce the frequency of maximum damping for any temperature, since this frequency is the reciprocal of the relaxation time. Thus at -15°C the maximum damping occurs at a little below 1 kHz. The value of the maximum logarithmic decrement, defined as the factor by which the amplitude A is decreased with time so that A(t) =i4(0)exp(- co td'/n), is 3.3x 1(T2 for the most highly damped case, i.e. transverse waves with wave-normal perpen­ dicular to the c-axis. In polycrystalline ice a further phenomenon occurs (Fig. 4). At high temperatures for a given frequency (or low frequencies for a given, fairly high temperature), the damping rises again, and continues rising without showing an apparent peak.80 This has been ascribed to grain-boundary friction by Kuroiwa,80 who has made a study of the effect in specially prepared samples with known boundaries. From this work he has been able to deduce an activation energy for grain-boundary fric­ tion of about 2.6 eV in pure ice. This effect will presumably exist whenever there are grain boundar­ ies in the ice, so that in this respect ice with all the c-axes aligned does not behave like single-crystal ice, whereas for other elastic and anelastic properties it does. The damping of seismic waves in ice masses is one aspect of this phenomenon, and in studies of such damping on the Athabasca Glacier18 it has been shown that, as expected, shear waves are damped more than longitudinal waves. It has also been possible to extend the data obtained in the laboratory80 on the grain-boundary damping of ice. The addition of impurities to ice affects the anelastic properties in several ways. First, it affects the movement of L-defects and other electrical point defects, and so alters the main anelastic relaxa­ tion peak. This phenomenon has been studied for NH4 F ,80 124 HF,^° 110 NaCl, HC1 and NaOH.80 THE MECHANICS OF ICE 9

\

Figure 4. Temperature dependence o f internal friction in pure poly crystalline ice. (After ref 80.)

The general effects are much as expected for the NH4F and HF in that the relaxation time is reduced and the activation energy affected. For NH4F doping124 two activation energies can be found; at high temperatures and low dopings Q has the same value as for pure ice, while for low temperatures and high dopings a Q of 0.1 eV was found. For HF80 110 the activation energy with large HF dop­ ings is about 0.25 eV, consistent with the activation energy for the movement of L-defects. Kuroiwa80 also reports a similar Q for the case of NH4F doping. Secondly, impurities can modify the grain-boundary friction damping. NaCl in increasing concen­ tration reduced the activation energy for grain-boundary viscosity from about 2.6 eV to about 1.3 eV. HF produces no such change in activation energy. The probable explanation of this effect is that the impurity concentrates in the grain boundary and gives greater disorder, or, near the melting point, leads to grain-boundary melting. Finally, the presence of impurities leads to a completely new anelastic absorption peak at a lower temperature than the peak for pure single crystals (Fig. 5). The effects for the different dopings are rather confusing; the simplest case is probably that of NaCl, where the peak is well separated from the pure ice peak. It occurs at about -150°C, and has quite a different frequency dependence. Un­ like the pure ice peak, its position does not vary markedly with frequency, but its magnitude does. The explanation of this phenomenon is not completely clear; Kuroiwa suggests that it may be due to vibrations of impurities trapped at local imperfections in the lattice. A study of anelastic behavior can give information about the state of impurities in a natural ice mass. Kuroiwa80 has studied samples of glacier ice from Greenland, Antarctica, and the LeConte Glacier in Canada. The main damping peak of the Greenland and Antarctic samples gave activation energies typical of those for impure ice, while the Canadian sample had an activation energy more like that of pure ice, despite the fact that its impurity content was quite appreciable. Kuroiwa at­ tributes this difference to the fact that the LeConte Glacier is temperate, and this will have allowed chemical impurities to diffuse to its grain boundaries while in the other samples the continued low temperatures have kept the impurities within ihe grains. 10 THE MECHANICS OF ICE

Figure 5. Temperature dependence of internal friction in NaCl-doped ice crystals. (After ref 80.)

CHAPTER III. PLASTIC DEFORMATION AND CREEP OF ICE - SINGLE CRYSTALS

Ice single crystals undergo plastic deformation very readily. This has been known for a long time, there being reports of the phenomenon in the last century.84 89 An ice crystal can slip readily on the basal planes, i.e. the planes perpendicular to the c-axis, and so provided there is a component of shear stress acting on these planes, plastic deformation can occur. The common view that ice is inherently a brittle substance which is hard to deform arises because most naturally occurring ice crystals are oriented in such a way that it is not very easy to put a shear stress on the basal plane. Thus a flat plate of ice from the top of freezing water normally has its c-axis perpendicular to the plate, and a bending stress applied to the plate consists mainly of tensile and compressive stresses perpendicular to the c-axis. Similarly a columnar ice crystal is most easy to grow with the c-axis either parallel or perpendicular to its long axis. In either case, a tensile stress on such a crystal will have no shear component on the basal planes. THE MECHANICS OF ICE 11

Attempts have been made to see whether there is any preferred direction for the deformation in the basal plane. All attempts to do this at temperatures near the melting point have yielded the result that no such preference exists,32 89 113 114 and only at very low temperatures34 can a prefer­ ence be found; this is in the < 1120> directions. The implication of this result is that at high tempera­ tures slip occurs within the basal plane in the direction of maximum shear stress, while at low temp­ eratures (about -60°C) the direction deviates from that of maximum shear stress towards the <1120> direction nearest to that stress. The theoretical explanation of this behavior is that ice deforms most readily by the movement of dislocations on the basal plane having the most likely Burgers vectors (see ref. 36, Chapter III), and that more than one slip system (i.e. more than one Burgers vector) is involved whenever the shear stress deviates from being parallel to the Burgers vectors. The way in which the different slip sys­ tems might take part has been discussed by Kamb,70 who showed that it was plausible for the slip in the basal plane to be apparently independent of direction. While the plastic deformation mechanism described above is by far the easiest to induce in ice, it is not the only one. By doing careful experiments on crystals unfavorably oriented for basal glide, the existence of plastic deformation on non-basal slip systems has been demonstrated in ten­ sile experiments,91 and the existence of such deformation has been deduced from other evi­ dence.5 82 90 102 The most likely dislocations in most of these cases have the same Burgers vectors as before, i.e. a/3 < 1120>, but the slip plane is no longer (0001) but either a prismatic plane |l0 l0 j or a pyramidal plane such as jlO ll} . There have, however, been reports of other Burgers vectors from indirect evidence. At present this whole subject is not well clarified, and only the strictly mechanical evidence will be discussed further. This arises from the tensile experiments already men­ tioned, which are consistent with the usual Burgers vector as for basal glide and the prismatic slip plane, and also from observations of cross-slip,5 where dislocations slipping on the basal plane have a small region where they cross slip from one such plane to another. This can occur when the dislo­ cation line is parallel to the Burgers vector (i.e. the dislocation is a pure screw dislocation) and neces­ sarily means that the region of non-basal slip has the same Burgers vector as the basal slip itself. The small sections of cross slip look as though they are perpendicular to the basal plane, and so are con­ sistent with the tensile tests in giving evidence for prismatic slip planes.

Creep of ice single crystals Plastic deformation is referred to as creep if it occurs under constant load or constant stress. Creep tests have been made on ice single crystals under a number of different stressing conditions, including tension, compression, bending, shear, and combined stresses (biaxial and triaxial tests). The general results of such tests, though differing in detail, are consistent with the view that, unless the orientation of the specimen is such that the shear stress on the basal plane is very low, the speci­ men deforms by slip on the basal plane, and the rate at which such slip occurs increases with time. This result is different from that found in, for example, metal single crystals, or indeed in polycrys­ talline ice (see Chapter IV), but is similar to that obtained with certain other nonmetallic crystals, notably some alkali halides and the semiconducting elements Si and Ge (it is of interest that these elements have structures similar to that of ice Ic). Typical creep curves are shown in Figure 6. It has been found34 54 68 that the curves for a particular kind of test are all of the same shape. The usual explanation of this accelerating creep is that initially there are a relatively small number of dislocations available for the deformation, and that their velocity is limited at the stress available. As deformation proceeds and more dislocations are produced by multiplication processes, the rate of deformation can accelerate,, this effect being more important than the interaction of dislocations which causes the reverse effect in metal single crystals. The shape of the creep curve has been 12 THE MECHANICS OF ICE

Figure 6. Creep curves o f ice single crystals deformed under vari­ ous stresses at —70° C. (After ref. 68.) approximated by a power law in which the strain e is proportional to the time t raised to some power m. In low temperature (below -60°C) tensile tests, and for small strains,34 68 the value of m is about 1.5; in compression and at rather higher temperatures44 it is about 2. At higher strains, there is some evidence that the rate increases less than predicted by the power law113 and tends to a more nearly steady rate. The stress variation of the creep rate has been represented by a power law in which the strain rate e at a given strain is proportional to stress o raised to the power n. The best values of n found by different experimenters have been 2 in compression,44 1.58 in bending,91 and 2.2 in tension.34 There is some evidence that the best value for n depends on the stress,34 being higher at larger stresses, and also some evidence113 that it may vary according to the stage of the creep curve at which it is calculated, being higher at the earlier stages. The temperature variation of the creep rate has usually been considered on the assumption that it is governed by an Arrhenius type relation

e= Kon exp(-Q/kT) where k is Boltzmann’s constant and the activation energy Q is 0.68 eV for temperatures between -50° and -10°C54 66 but is 0.41 eV for temperatures between -90° and -60°.C. These values are sufficiently near to the values for electrical and anelastic relaxation in the high temperature range for the motion of dislocations to be governed by the movement of electrical point defects as sug­ gested in Chapter III of ref. 36. The reduction at low temperatures could be due either to the rela­ tive importance there of defects contributed by impurity atoms (which do not require a formation activation energy of the same magnitude), or to the change in the nature of the defect responsible for most easy relaxation from the L-defect to the positive ion, a change also suggested by other data as discussed in ref. 36, Chapter V. Another phenomenon which has been proposed as responsible for determining the creep rate of ice is the so-called Eshelby-Schoek viscous dislocation damping. As we saw in Chapter II, the orien­ tation of water molecules is probably not random in the presence of stress; this reorientation under the varying stress was thought to be responsible for the damping of elastic waves in ice. The stress field around a dislocation will similarly produce a situation in which water molecules have equilibrium THE MECHANICS OF ICE 13

Figure 7. Stress-strain curves o f ice single crystals obtained at a strain rate o f 2.7x1 O'1 s~l , and at various temperatures. (After ref 68.) orientations not quite at random, and when the dislocation moves it should carry this pattern of water molecule orientation with it. It can only do so, however, if the water molecules are free to rotate to speeds faster than necessary to keep up with the dislocation, and clearly this will not al­ ways be the case. Weertman128 has developed a theory based on this phenomenon, and shows that it predicts a value for the creep rate of the right order of magnitude and a stress dependence of a power law with m = 3. The temperature dependence would of course be the same as with any other mechanism depending on the same process for reorienting, i.e. the same as for dielectric and anelas- tic relaxation, and as discussed in the last paragraph the experimental data are not inconsistent with that assumption.

Constant strain-rate tests If instead of applying a constant load or a constant stress, an ice crystal is deformed in apparatus which ensures a constant rate of deformation and measures the stress, a conventional stress-strain curve is obtained.53 66 68 101 Since the strain-rate increases in a creep curve at constant stress, we would expect the stress needed to keep the strain-rate constant to fall with time, and this is what is observed. In fact most constant rate curves start with an increasing stress, but this is due to stress­ ing of the machine, or to elastic defects on the specimen, and is not really representative of the plastic deformation of the ice. A typical set of constant strain-rate curves is shown in Figure 7. The maximum in the curve is rather like the yield point in an iron specimen, but the physical explanation is quite different. It nevertheless is often referred to as a yield phenomenon. As has already been sug­ gested above, the explanation in this case is just the same as the explanation for the accelerating creep curves - that dislocations are multiplying and so making it easier for the ice to deform; the larger number of dislocations at a higher strain do not have to move so fast in order to keep the overall strain-rate constant, and so, since stress is related to dislocation velocity, the stress required is not so large. The theory of stress-strain curves based on this mechanism has been adapted for ice53 68 101 and can be made to fit the observed curves for reasonable values of the parameters. The agreement is not as convincing as in the case of IiF, for which experimental observations of dislocation veloci­ ties under a known stress are available, but observations of dislocation velocity in ice30 in a situation where the stress is not quite certain (a bending test), of 1CT5 m s '1 at -22°C, is reasonably consistent. 14 77//: MECHANICS OF ICE

Figure 8. Stress-strain curves for non-basal glide o f ice single crystals with various orientations o f the a- axis. The tensile axis lies in the basal plane. (After ref 55.)

Plastic deformation of ice crystals in non-basal glide As has been mentioned above, ice crystals can be stressed under conditions in which dislocations on the basal plane are not capable of flowing. In particular, this happens in a tensile test if the ten­ sile axis is parallel or perpendicular to the c-axis. Dislocations with Burgers vectors lying in the basal plane but capable of gliding on other planes can produce deformation if the tensile axis is per­ pendicular to the oaxis, but not if it is parallel. There are thus two rather different cases of “hard glide ’ in tension. The first case (c-axis perpendicular to the tensile axis) has been studied in some detail,91 and it is found that plastic deformation does occur in this case, but that the stress required to produce a given strain-rate is very much higher than for basal glide. The shape of the creep curve is also quite different. Figure 8 shows examples of non-basal stress-strain curves, which are much more like those in metal single crystals, and also, possibly significantly, more like those of polycrys­ talline ice. It has also been shown in shear tests16 that the stress required for non-basal glide is much larger, and is more like that for polycrystalline ice. The stress variation is still a power law16 with a value oi n of about 2.7, or, according to the tensile data,55 about 7. The activation energy for this creep is about 0.52 eV. During non-basal creep, voids are observed to form in the ice, provided the experiment is per­ formed in tension.83 91 These voids are probably formed from vacancies generated as the disloca­ tions move. The theory of this process has been worked out83 and agrees reasonably well with the experimental observations. This agreement, which involves taking the activation energy for vacancy diffusion as 0.67 eV, is further evidence for vacancy diffusion as the mechanism for self diffusion in ice. '

Effect of surface conditions on plastic flow of ice crystals it has been found that the actual magnitude of the maximum stress in constant strain-rate tests is affected quite markedly by the surface of the ice.92 Chemically polished ice crystals have a much higher yield point by a factor of about two than mechanically polished samples. The tempera­ ture variation of the maximum yield stress is of the same form in both cases, however, indicating THE MECHANICS OF ICE 15 that the same mechanism is responsible (the activation energy is the same in both cases, and is dif­ ferent from that for similarly treated polycrystalline columnar ice). The probable explanation for this is that the mechanical polishing introduces surface dislocations which can take part in the plastic deformation. This explanation is in agreement with the observation that a long period of annealing just below the melting point restores the yield point of mechanically polished specimens almost to that of the chemically polished samples. It is also consistent with the observation that crystals which have sub-boundaries within them (i.e. which already have walls of dislocations) have low yield stresses that are not much affected by the mode of surface preparation.

Effect of an electric field on plastic flow of ice crystals If dislocations in ice are charged, as will be the case if the number of dangling bonds with and without protons are not equal, then it is to be expected that an applied electric field would exert a force on dislocations in ice, and so might affect plastic deformation. However ice does have a finite electrical conductivity, so it is difficult to apply a continuous electric field within ice, particularly since the mechanism of electrical conduction is believed to be by ionic defect migration (see Chap­ ter V of ref. 36), and therefore even if electrodes are put in contact with the ice, ionic migration will tend to remove the field within the specimen unless the electrodes are of a kind that can receive protons and so release the ions. If an alternating potential is applied to ice of sufficiently high fre­ quency, we would expect the field to penetrate into the ice, and hence would expect dislocations to experience an alternating force, and this might add to the mechanically applied force and help dislocations to overcome barriers and so enhance the creep rate. For an appreciable part of the field to penetrate into the ice, however, the frequency needs to be of the order of magnitude of the re­ ciprocal of the relaxation time for the conduction process. However, despite this theoretical predic­ tion, relatively low-frequency a-c fields applied to ice59 (such as that of the normal 60-Hz electricity supply) have been reported as producing observable movements of dislocations in ice as seen by X- ray topography. The movement of dislocations within a crystal whose conductivity would seem to be too high to allow the field to penetrate has also been reported in alkali halide crystals; the phen­ omena are discussed in a recent review by Whitworth.131 A 60-Hz a-c electric field of 600 V cm“1 has also been reported as increasing the creep rate of an ice single crystal,58 and this could be due to the action of the alternating force, although the possibility of its being due to slight heating ef­ fects or to changes in the population of electrical defects in the ice should also be considered.

Effect of impurities on the flow of ice single crystals The main impurities which have been found to have large effects on the mechanical properties of ice single crystals are the acids HF and HC1. The effect is particularly marked at low temperatures.67 As can be seen from Figure 9, one part per million of HF dissolved in ice is enough to reduce the yield stress to about one-third, and the same sort of concentration increases the creep rate at a con­ stant stress by one or two orders of magnitude at -70°C. The activation energy is also affected; al­ though the accuracy of the experiments is not high, it seems to be about halved as compared with the higher temperature range for pure ice crystals, i.e. 0.33 eV compared with 0.68 eV. This means that the difference between HF-doped ice and pure ice becomes progressively less as the temperature rises. The probable explanation of this phenomenon35 is that the plastic deformation is controlled by the necessity to reorient hydrogen bonds in front of the moving dislocation (see Chapter III of ref. 36), and that this occurs by the migration of electrical point defects. HF introduces such defects, and while the number introduced by HF is large compared with the number occurring naturally in pure ice due to thermal activation, there will be a large difference in behavior between HF-doped and pure crystals. At higher temperatures, the number of naturally occurring point defects is greater and 16 THE MECHANICS OF ICE

a.

Figure 9. Creep curves (upper diagram) and stress-strain curves (lower diagram) o f ice single crystals a t -70° C with various con­ centrations o f HF, compared with pure specimens. (After ref 67.) the effect correspondingly smaller. However it is such a large effect that it remains observable to high temperatures. Other dissolved impurities which introduce point defects could be expected to have similar ef­ fects, and indeed HC1 has been found to do this.94 The main difference between this phenomenon and that for HF is that it appears to be independent of the concentration of HC1, while the effect on strain-rate was approximately proportional to the square-root of HF concentration.67 This sug­ gests that the amount of HC1 strictly in solution in the ice is not varying, and that the greater amounts of HC1 determined by analysis are not available for release of defects. This is consistent with what is known about the relative solubility of HF and HC1 in ice.134 It might also be expected that NH3 and NH4OH, which dissolve in ice, might produce effects as well. However tests indicate67 that NH3 produces if anything a slight hardening of the ice, i.e. an effect in the opposite direction from that produced by HF and HC1, while NH40H produces no marked effect at all. This must indicate either that these do not release defects (in the case of NH3 presumably D-defects), or else that these defects are not able to perform the necessary reorientation around the dislocation.35 THE MECHANICS OF ICE 17

Figure 10. A bent ice crystal viewed by shadow photography showing the slip bands. (After ref. 95.)

Slip lines and slip bands in ice Throughout the discussion above, we have talked about slip on particular glide planes without discussing exactly how these planes are located in the ice crystal. As with other materials deform­ ing by dislocation movement, it is found that the deformation is not uniform on a microscopic scale, but that the slip is concentrated into particular slip planes, or into groups of slip planes very close to one another. In particular, for basal slip in ice the active slip planes may be relatively widely spaced. This is not altogether surprising since, as we have seen, some features of the plastic defor­ mation can best be explained by assuming that in ice there are relatively few dislocations taking part in the plastic deformation initially, and that the subsequent acceleration of the creep (or reduc­ tion of stress in a constant strain-rate test) is due to multiplication of the number of dislocations from the initial few. Most processes of dislocation multiplication, such as the action of Frank-Read sources or cross-slip of dislocations, produce further dislocations either on the same slip plane or on a close neighbor and thus should cause the slip to be concentrated into particular planes or bands. These bands can be seen in deformed ice crystals using Schlieren techniques, and the resulting linear structures can be quite dramatic, for example in a bent ice crystal95 (Fig. 10). The slip line spacing has been studied;102 121 the spacing of visible slip bands d decreases as the stress a increases above about 0.2 bar according to the relation

(a - oQ)d = k

where a0 = 0.2 bar and k = 4.5 /un-bar. This relation would imply that at a stress of 0.2 bar the slip-line spacing would become infinite and there should be no slip, i.e. that this is a fundamental yield point for ice; however, below this level fine slip bands of a rather different appearance are still observed.102 These have a much closer spacing which fits with the relation

od = k' 18 THE MECHANICS OF ICE where k' = 0.72 ¿im-bar, a value which fits well an old fundamental formula of dislocation theory according to which k' = 0.16 Ga where G is the shear modulus and a the lattice constant. Optical phenomena associated with slip bands have also been studied.5 The presence of effects in polarized light is interpreted as evidence for some non-basal glide involved in these essentially basal slip zones.

Effect of stress perpendicular to the slip plane on basal slip of ice crystals In the simple theory of plastic deformation by dislocation movement it is the resolved shear stress on the slip plane in the slip direction which alone is responsible for determining the plastic deforma­ tion caused by the dislocations concerned. In ice there are three possible directions for Burgers vec­ tors in the basal plane, and Kamb70 has shown that, provided these different sets of dislocations be­ have quite independently of each other, the flow will essentially be in the direction of maximum shear stress in the basal plane, so that this component will be the important one for determining the flow of the crystal. Strictly this result holds if the value of the exponent n in the power law for the dependence of creep rate on stress is equal to 1 or 3. If, as seems to be the case (see above), m lies between 1 and 3, then the flow will be slightly more rapid if the stress is halfway between two slip directions (i.e. in a < 1010> direction), while above 3 it will be more rapid if the stress is in a < 1120> direction. However for 1 < n < 4 the effects would not be strong enough to be readily detectable by existing experiments. There has been some testing of this hypothesis. A compressive stress applied perpendicular to the slip plane ought not to influence the flow of the ice at all, according to this theory. It has been found, however,114 that such a pressure did increase the flow velocity according to the relation

ln e = 7 (P-P0) where e is the creep rate, p is the pressure, and y and p0 are constants. This relation has the start­ ling implication that at p = pQ the material will strain with no shear stress being applied at all. It is possible that this effect is due to accidental misorienting of the specimen so that at p = p0 the accidental shear stress on the basal plane due to this misorientation is equal to the shear stress necessary to flow at the specified rate; the values of p0 were quite large, ranging from about 10 bars at -5.7°C to 360 bars at -21.7°C. Perhaps a more significant experiment is done by superpos­ ing a hydrostatic pressure onto the shear stress. In this case no accidental misaligning can cause stray resolved shear stresses. In an experiment of this sort,104 it has been found that no marked ef­ fect occurs provided the temperature of the specimen is adjusted to take account of the change of melting point caused by the applied pressure, i.e. provided the temperature is the same number of degrees below the melting point of the sample.

Stress relaxation in ice crystals Since ice undergoes creep at a constant stress, it follows that a specimen which is held at a con­ stant length will slowly relax the stress necessary to keep it at this length. This phenomenon is called stress relaxation, and has been studied in ice crystals.101 121 Comparison of the experiments of Wakahama121 with others discussed in this chapter is difficult because he did not observe the yield phenomenon in his constant strain-rate tests (the reason for this is not clear). His relaxation curves followed the equation

o THE MECHANICS OF ICE 19 where o0 and A are constants, a formula he was able to explain on the basis of a simple dislocation theory, in which it was assumed that the velocity had a particular relation with the shear stress on a dislocation deduced from the flow curves. If the creep curve has the form € = kon, then the stress relaxation should have the form a = - akon where a is a constant depending on the geometry and elasticity of the specimen. This formula is consistent with Wakahama’s if n = 2. As we have seen above, creep curves give values of n which are close to 2, and constant rate curves101 give a value of n which appears to vary somewhat with strain but which is of order 2 with considerable scatter. The various results are therefore in substantial agreement within their experimental accuracies.

Hardness of ice crystals Brinell indentation tests and scratch hardness tests have been reported for ice single crystals.12 73 Because of the extreme anisotropy of the plastic properties of ice, with its single set of planes (those parallel to the basal plane) being much easier to glide on than others, it is hardly surprising that hardness is different according to which orientation the ice crystal is in. At temperatures below -10°C, indentation tests parallel to the c-axis (i.e. into the basal planes) and scratches on planes containing the c-axis (i.e. on prismatic planes) gave higher hardness values.12 This is not altogether unexpected, since these are deformations that are difficult to accomplish by basal glide alone, where­ as indentation into prismatic planes, or scratching of basal planes, could occur by such a mechanism. What is less easy to understand is that this difference appears to be strongly temperature-dependent and to reverse in sign above -10°C. This interesting phenomenon deserves further study ; it has not yet been satisfactorily explained.

CHAPTER IV. PLASTIC DEFORMATION AND CREEP OF POLYCRYSTALLINE ICE

Ice poly crystals deform much more slowly under otherwise similar conditions than do single crystals. This is not hard to understand if we consider what must go on between the individual crystals within poly crystalline ice. Each individual grain can deform readily on its own basal planes, but since these are in different directions in different crystals, individual crystals will oppose the movements in their neighbors. It can be shown that, in order that a crystal be able to deform to any desired shape compatible with maintaining its volume, it must be capable of deformation on five independent slip systems. The basal slip plane of ice, even though it has three slip directions, has only two such independent systems (this is because slip on one of the three systems can be made up by slip on the other two. Thus if. the only way in which ice could deform were basal slip, deformation of polycrystalline ice would be impossible. In fact we would expect some other process to occur. Cracks might open between the grains, or grain-boundary sliding might occur and mate­ rial might diffuse along the grain boundaries to fill up the volumes between the moving grains. This latter process is indeed believed to happen, particularly at temperatures close to the melting point (Barnes et al.4 believe this happens between -8° and -1°C). At still higher temperatures the regela­ tion process may help grains to deform, so that ice at points of high pressure between grains melts and flows along grain boundaries to points of lower pressure.3 4 However at lower temperatures neither of these processes is thought to be dominant, and, as we have seen in Chapter III, ice is cap­ able of slip on non-basal systems, although their exact identity has not been fully established. Thus except close to the melting point nonbasal slip may well be the controlling process governing plastic deformation on polycrystalline ice.115 The case for this being so is stronger because of similarities in the general shape of the creep curves of single crystals deforming by non-basal slip and those of polycrystalline ice, as well as ttnUaiittadncihe stress level in these two cases.16 But if the non-basal slip still has only dislocations whose Burgers vectors lie in the basal plane, as would 20 THE MECHANICS OF ICE be the case for example if they are generated by cross slip, then this plus the basal slip still only amounts to four independent slip systems and ice still could not deform to a general shape in any crystal orientation. To see this, consider a single crystal within a poly crystalline mass which is being compressed in uniaxial compression, and for which the oaxis is parallel to the compression axis. In simple theory, such a crystal has no resolved shear stress on its basal plane, no resolved shear stress on prismatic planes, and the resolved shear stress on pyramidal planes is perpendicular to the intersection of those planes with the basal planes, and hence to the Burgers vectors we are hypothe­ sizing. However the presence of a small number of such essentially rigid grains within the aggregate would not stop deformation completely. Their presence is surely much less serious than the situa­ tion where all grains with their oaxes perpendicular to the stress are similarly rigid, since in a ran­ domly oriented poly crystalline aggregate there will be many more of such grains around a girdle on a petrofabric diagram. From experimental evidence it also seems that their presence is not of great significance, since the two main kinds of ice used in laboratory work give results that are broadly similar. These two kinds are 1) randomly oriented polycrystals, and 2) ice with columnar grains with their c-axes approximately perpendicular ta the column axes. The method of preparation of ice samples with these characteristics has been discussed in Chapter IV of ref. 36. Since the second kind has a much more restricted range of c-axis orientations, among which is the orientation with c- axis parallel to the uniaxial stress, which is normally applied perpendicular to the column axes, this sort of ice should have many more grains in the more nearly rigid orientation. It may, of course, be significant that observations of detailed mechanisms of ice deformation in this kind of ice show a large number of cracks developing. However, since a similar investigation has not been made for randomly oriented ice, it would be unwise to assume that this does not occur in that case as well. The third type of polycrystalline ice discussed in Chapter IV of ref. 36 has all its c-axes parallel, and thus all the basal planes parallel. It should therefore be as anisotropic in its plastic properties as a single crystal, though it does not follow from this that it should behave exactly as a single crys­ tal would, because the Burgers vectors in the different crystals will be in different directions. Not much work has been done on such ice; its properties might be of interest in understanding the role of grain boundaries in inhibiting slip from one grain to another.

Results of creep tests on poly crystalline ice Creep tests on polycrystalline ice have been made by many workers. The most commonly used stressing systems are compression,* tension,10 60 113 and shear,16 100 105 although other systems such as hydrostatic pressure on a hollow cylinder,52 62 indentation hardness,3 bending23 75 and ex­ trusion65 have been used, and some experiments have been done to test the importance of different stressing systems on creep of ice in triaxial or high pressure tests.45 113 117 119 On the whole these experiments are qualitatively consistent, although there are exceptions, and the quantitative dis­ crepancies can usually be accounted for by, or attributed to, peculiarly difficult experimental con­ ditions, such as occur in attempts to work very close to the melting point. Rather than list in detail the results of these many experiments, we shall here discuss the general features of importance, giv­ ing reference to papers where the matter under discussion has been well studied, or where a detailed discussion of the different data can be found.

Shape of the creep curve Unlike single ice crystals, polycrystalline ice normally shows decelerating creep, at least initially. Provided the stress is low (the actual value depends on temperature; it is typically less than a few bars) the curve seems to decrease continuously. Typical curves for different specimens under the

♦Refs. 16, 17,22,31,41,46, 86, 87, 88, 113. THE MECHANICS OF ICE 21

TIME, doy*

Figure 11. Low-stress creep curves for fine-grained ice whose crystal orientation was initially random. Test temperature -2.06cC. Constant axial compressive stress o f 0.42 bar on ice which had not been previously loaded. (After ref. 87.) same conditions are shown in Figure 11. An exception to this is reported in columnar ice,41 75 for which a rate maximum at a strain of 0.25% is found. Possibly this is a reflection of the fact that this ice is nearer to single crystal ice than the randomly oriented ice of the other studies. The shape of the creep curve has been studied, and attempts made to fit an analytic form to it. The most usual form tried has been the power law where strain e is proportional to time t to some power p. This is a form commonly found for other materials for which, at high temperatures, p is often V3. In ice this has been found to give quite a good fit,31 while other work41 60 prefers a value nearer to %. At higher stresses (Fig. 12) a linear term has to be addedto this power law creep,31 and it seems Ekely that at all stresses, if the test were continued long enough, a steady-state rate would be reached. At still higher stresses a reacceleration is found31 113 which is attributed to recrystallization. When this occurs there may be a further deceleration, or there may be a new, higher steady rate established.65 All of this makes the creep of ice quite a complicated phenomenon, and if it is desired to predict how ice will behave in any situation, all these different phenomena must be borne in mind. If the stress continues to be applied to the ice for some long time, as in a glacier, then it is the final steady state to which the creep settles down that is probably of most interest, and many of the papers on the subject have had as their aim the establishment of some flow law usable for such purposes. A flow law of this kind may not be appropriate for predicting how ice will react on some other time scale, for example how an ice runway will deform under a parked vehicle, or how a newly opened borehole will close. Transient creep will normally make such flow faster than that predicted from a steady-state flow law, and prediction in these circumstances js much more difficult. The situation is made worse by the fact that the stress dependence of the different sorts of creep seems to be dif­ ferent. Some attempts have been made to produce more general prediction equations,41 60 but pre­ dictions must still be treated with caution. 22 THE MECHANICS OF ICE

Figure 12. Creep curves o f randomly oriented poly cry stal­ line ice (above) at -0.02°C under various stresses (below) at various temperatures under 6 bars. (After ref 31.)

The flow law variation of strain rate with stress The most commonly sought clarification of the creep of ice is a flow law which will relate creep rate to stress and temperature. As has been indicated in the previous section, this is a simplification, that may for many problems be an oversimplification. Even for the situations where transient creep is neglected, there are two possible ways of trying to identify a flow law, and there is some confusion in the literature as to which of the two is being sought in some cases. The first flow law, and for practical application probably the most useful, is that relating stress to the strain-rate which is estab­ lished at very long times under the action of that stress. At high stresses this rate may be higher than thé minimum creep rate, because of reacceleration due to recrystallization. As is discussed in Chap­ ter VI, this recrystallization may produce a grain size and crystal orientation pattern quite different from that which the ice had when it started, and it should therefore be borne in mind that ice in this state may no longer be isotropic. The second form of flow law is one which relates the secondary or steady-state creep rate to the stress, i.e. which relates the rate before recrystallizationto stress. Since recrystallization normally results in an increased flow rate (either because the recrystallized grains are “new” and therefore soft, or because they are more favorably oriented for flow, or both), this second flow law will predict slower deformation at high stresses than the first* It is of more significance from a fundamental physical point of view, since it is a law which relates to the defor­ mation of similar ice for all stresses. The distinction between the two rates is probably made most clearly by Steinemann.113 THE MECHANICS OF ICE 23

The most usual analytic form suggested for the stress variation is a power law

e = A on where e is the creep rate, o the stress, and A and n are constants. While n is dimensionless,^ has dimensions (time)(stress)“'\ This rather awkward situation leads some to use the law in the form

. e/e0 = (ct/ct0)" where eQ and oQ are constants with the dimensions of strain-rate and stress respectively, but this gives the false impression that three parameters are needed to specify the law, whereas only two are required. This power law is often associated with the name of the present author, because he first suggested its use31 and showed that for the second (minimum creep rate) kind of flow law it fitted data from 1 to 10 bars, and that hardness data implied its approximate validity to much higher stresses, a result confirmed by more recent hardness data.3 The value of n used by Glen for the steady-state creep (i.e. allowing for the further deceleration in low stress tests) was 4.2; if minimum observed creep rate was used with no allowance for further deceleration, the value obtained was 3.2. The value of n found for hardness tests, where the stresses were 10 to 100 times greater, was 3.85. Most subsequent tests within the same sort of stress range, and for which tests at lower stresses were continued long enough, give similar values between 2 and 4,16 22 41 113 and it is reported that D20 ice gives a similar value.129 This value also seems to fit data from measurements on glaciers, although here the problem of identifying the stress acting on the ice is fairly acute. There is some suggestion113 that the value of n is rising slowly through the region from 1 to 15 bars, though if the hardness data are to be believed this rise cannot continue much above this level. The other flow law, the one relating the final flow rate including any recrystallization, has not received so much study. It has also been approximated by a power law;113 it starts at stresses up to 12 bars by being rather similar to the previous law, but at high temperatures and stresses the ap­ parent power rises rapidly to about 10. At lower stresses the problem that transient creep has not finished becomes very acute. Most workers have ignored this and plotted their minimum rates, or the rate for a given strain or for a given time. The result of this is to change the slope of the power law to approximately unity, i.e. at low stresses ice appears to be behaving like a viscous material.10 17 This interpretation has how­ ever been challenged,87 and in tests made by first stressing at a higher stress and then reducing (a phenomenon we shall return to below, since it has some analogies with strain relaxation), it has been shown that a more likely relation is a power law with «=1.8 between 0.1 and 0.5 bar, a value which also has some justification from glacier data56 (Fig. 13). Probably the best assumption to be made at present is that the relation between stress and strain rate approximates to a power law with n = 3.5 at stresses above about 1 bar, and that the ap­ parent value of n falls off below this stress level towards (but not necessarily reaching) unity. Such a flow law is compatible with the data, and is not ridiculous theoretically. If we knew more cer­ tainly just what was responsible for controlling the flow rate of ice, it might be possible to make more reliable theoretical arguments concerning the flow law. Weertman129 has discussed the form the creep law might take; at high stresses ice might have creep dominated by dislocation climb or by microcreep processes. At very low stresses Nabarro-Herring creep may become important. This is creep produced without the slip of dislocations by vacancy or interstitial diffusion through the individual crystals. It cannot produce creep of the magnitude that is readily observed in the lab­ oratory except with very small grain sizes, but it may nevertheless be important in producing very 24 THE MECHANICS OF ICE

Figure 13. Axial strain-rate as a function o f axial compres­ sive stress for fine-grained, randomly oriented ice at -2.06° C. The points marked by circles are from ref 87, that marked A from ref 86, and that marked B from ref 88. The slope at the high stress end corresponds to that established in ref. 86. (After ref. 87.) slow creep under low stresses in glaciers and ice sheets. It is Newtonian viscous (i.e. strain-rate is proportional to stress) provided the grain size remains constant with time. Bromer and Kingery10 suggested it might be responsible for the linear variation of strain-rate with stress they observed in the laboratory but although their stresses were in the range where we might expect the effect to be important, their observed strain-rates were much too high, being some 100 times higher than would be predicted from the known self-diffusion coefficient of ice. Their strain-rates were also much larger than those found by others for poly crystalline ice, and since they were using columnar ice with all c-axes subparallel to the column axes, this is perhaps not surprising. Their tests were not continued long enough to ensure proper steady-state conditions, however, and it is doubtful if theirs is a real observation of Nabarro-Herring creep in ice.

Variation of creep rate with temperature The creep rate varies quite rapidly with temperature. Most authors have analyzed this variation by assuming that we are dealing with a thermally activated process for which the Arrhenius relation holds, i.e. that

e ex exp(-Q/kT) where Q is commonly referred to as the activation energy (more accurately it is an activation enthalpy), k is Boltzmann’s constant and T is the absolute temperature. There are two ways of expressing this THE MECHANICS OF ICE 25

TEMPERATURE,°C

Figure 14. Logarithm o f strain-rate as a function o f 1/T for polycrystalline ice, also for single crystals compressed perpen­ dicular to their c-axes. (After ref. 88.) activation energy, either in molecular units (eV) or in macroscopic units (usually kcal/mol in the literature, but with the growing disfavor for the calorie more recently in the SI units of kJ/mol). Most of the literature is in kcal/mol. In this report, viewing the subject from the more physical point of view, we have used eV. The conversion is 1 eV = 23.0 kcal/mol = 96.2 kJ/mol. Actually, relatively few experiments have covered a sufficient temperature range to test the ac­ curacy of this expression, and those that have (Fig. 14) show that it is only true at lower tempera­ tures, below about -10 C.88 In this low temperature range, the value of Q has been reported as 0.72 eV (data from -10° to -60°C) although earlier experiments86 had given the lower value of 0.52 eV based on data from -0.5° to -35°C. These values are not inconsistent with the idea that self diffusion may be responsible for the rate process governing creep, as would be true on a dislo­ cation climb model, nor with the idea that migrating electrical point defects may be responsible as suggested also for basal glide in single crystals. There are no data to show whether the change in activation energy found in single crystals below -60°C also occurs in poly crystals. Of course, since polycrystals are much harder, the strain-rates involved are much smaller, and as we have already seen yet another process, Nabarro-Herring creep, may enter in polycrystals. At the high temperature end of the range, where the Arrhenius equation breaks down, it may well be that different phenomena are at work. Barnes and Tabor have suggested that pressure melt­ ing between the grains may be responsible for deviations from the Arrhenius law3 at the highest 26 THE MECHANICS OF ICE stresses, and it is certainly possible that some process involving the grain boundaries is involved. It could, however, be boundary slip and migration. Values of activation energies deduced at these temperatures frequently are very much larger than those reported above. Glen31 first reported a value of 1.4 eV from data in the range -1.4° to -13°C, and very high values have also been reported by Higashi (1.64 eV),52and, based on their hardness measurements, Barnes and Tabor (1.3 eV).3 Voytkovskiy117 118 120 prefers in this region a formula first introduced empirically by Royen

1 + 0 where d is the temperature difference from the melting point, i.e. the Celsius temperature without the minus sign. Steinemann113 also reported a failure to get a single activation energy, and reported values between 0.9 and 1.8 eV. In the light of these results it would seem unwise to use the Arrhen­ ius law as a formula for variation of flow law with temperature above -10°C. As well as references already quoted, other authors* have reported activation energies with values varying from 0.44 to 0.70 eV. Very close to the melting point, i.e. under the conditions prevailing in glacier ice, the flow law can be rather different. Indeed it is surprising that flow laws derived from tests at lower tempera­ tures have been so successful in the theory of temperate glaciers. In itself this indicates that the water which presumably is present in temperate glaciers is not playing too profound a role in the deformation of temperate ice. Steinemann113 made calculations to see where such water would be located from this point of view, but his calculations are not valid because they used inaccurate data for the surface energy of the ice/water interface. More recent data (see Chapter X of ref. 36) indi­ cate that instead of being concentrated in pockets where four grains meet, the water will spread out along lines where three grains meet. However, even here the effect on mechanical properties will not be so profound as it would be if water spread all over the grain boundaries (i.e. the surfaces where two grains meet). This thermodynamic result probably makes less likely the possibility of pressure melting on grain boundaries postulated by Barnes and Tabor,3 while not of course elimin­ ating it as a possibility in a rapid state far from thermal equilibrium. Attempts to produce and test temperate ice in the laboratory have been unsuccessful, even the giant ice viscometer9 being probably too small to escape the difficulties of thermal control of a small specimen. One attempt has been made19 to use a glacier itself as the thermal control. This experiment showed that the flow law for temperate ice is of the same general kind as has been dis­ cussed for ice at lower temperatures, being best described as the sum of a power law and a linear law.

Constant strain-rate tests Just as in the case of single crystals, it is possible to do a test at constant strain-rate and to meas­ ure the stress as a function of time or strain. Since the creep curve is now decelerating, we would expect the stress-strain curve so produced to be increasing, and this is what is found, the reaccelera­ tion associated with recrystallization now appearing as a maximum in the curve.22 46 The curves should settle down to a steady stress, just as the creep curves should settle down to a steady strain- rate, if the assumptions on which a flow law is based are justified.

Stress relaxation If the length of an ice specimen is held constant, the stress required to maintain this state will decrease with time. This is the phenomenon of stress relaxation, and it has been studied by

♦Refs. 10, 17, 22, 60, 62, 100, 129. THE MECHANICS OF ICE 27

Voytkovskiy.117 118 He has found that the amount of relaxation and the time for it to develop, as well as depending on the ice structure, temperature, and stress, depend on the time for which the creep had been occurring prior to the beginning of relaxation. Thus, for example,117 ice relax­ ing from an initial stress of 7 bars increased the time of half relaxation from 0.10 hour to 1.7 hours as the time of deformation increased from 0.1 hour to 8 hours. An analogous phenomenon occurs if the stress is reduced instead of being removed completely.31 87 If the stress is markedly reduced, a negative creep rate may be observed for some time before the strain reaches a minimum and then begins to increase again. This phenomenon has been used87 to try to set a lower limit on the asymp­ totic flow rate at low stresses, just as the normal creep curve sets an upper limit. Whether this is a valid way to find true steady-state creep rates at low stresses has been questioned.130 Stress relaxa­ tion itself has been suggested as a method for finding a complete flow curve in one test.120 This seems to suffer from the same objections, only more acutely; the transient phenomena are undoubt­ edly taking place during stress relaxation. Transient phenomena are more complicated than would be necessary if they were to be used in this way; when the stress is reduced considerably the creep rate may temporarily reverse in sign and then accelerate again before finally returning to a decelerat­ ing curve, or, if the stress is reduced by a small amount, these earlier effects may be swamped by the on-going, decelerating transient creep. It is not true that the creep rate always approaches monotonically the final “steady-state” creep rate appropriate for the stress. There are various possible reasons for these phenomena, including anelastic effects and recrystallization, but these have not been fully investigated.

Strain relaxation When the load is removed from an ice specimen, it does not remain at constant length; instead a slow negative creep takes place. This phenomenon can be regarded as anelasticity, and so might perhaps have been treated in Chapter II. However, it is probably quite different in origin from the damping peaks which occur in high frequency acoustic waves in ice, and is probably due to move­ ment of dislocations under the action of internal stresses in the ice; it is therefore more related to the forward creep discussed in this chapter. This phenomenon has been discussed by Krausz76 77 78 who has investigated the phenomenon experimentally and has also developed a theory based on the concept of two different activation energies, one forward and one backward. His original paper77 used single crystal data of Readey and Kingery (see Chapter III), and he then found the two barriers to be equal in activation energy, but his own experimental data allow the difference to be investi­ gated.

Effect of different stressing systems So far we have discussed the flow law of ice without considering the difference between tensile, compressive and shear stresses, nor the question of whether hydrostatic pressure affects the flow curve. In other words we have been ignoring the fact that stress is a second rank tensor quantity. The fact that the different tests do give roughly comparable results shows some kind of interrela­ tion exists, but it is obviously necessary to look at this in more detail. Ideally one could tell what the relation was by comparing the results of, say, compression, tension and shear tests, but this is not in fact really possible. The very different types of test specimens required for the different tests make it hard to get results that can be confidently compared. It is better to arrange for some apparatus which can take similar specimens and subject them to stress systems with differing amounts of the different stress components. This, however, makes testing much harder, and to date it has only been attempted by a limited number of workers.45 113 117 119 The results of these experiments are not altogether in agreement. Glen33 has discussed what is involved in these 28 THE MECHANICS OF ICE discussions at greater length than will be possible here. The problem can be reduced to two main questions: does the hydrostatic pressure affect the flow curve, and does the third invariant of stress affect the flow curve? Let us consider first hydrostatic pressure. Steinemann113 has made tests which, within their experimental accuracy, show it does not affect the flow curve to any marked extent; certainly a hydrostatic pressure of the same magnitude as the flow stress itself is insignificant. This result is comparable with that found for single crystals (Chapter III) where, it will be remembered, the flow stress was independent of pressure provided the temperature was kept the same relative to the melt­ ing point. This relation, reminiscent of the Royen formula, has itself been questioned in later work. Vyalov reports119 that flow rate is reduced if a pressure is applied. However, his pressure is not strictly hydrostatic, and the effect may be due to the third invariant o f stress. Haefeli and others,45 in what seem to be better defined conditions of hydrostatic pressure, find on the other hand that the flow rate is increased by a hydrostatic pressure at constant temperature, but decreased by a pressure at constant difference from the melting point. In other words, lowering the temperature to compensate for the lowered pressure melting point decreases the creep rate too much to compen­ sate the increase that has occurred. At present it is probably correct to assume that pressure does not have a very marked effect on flow rate, the slight effect being given in sign by these last men­ tioned results.45 The effect o f the third invariant of the stress tensor is a still more complex thing to investigate. To see what is involved, let us ask the question: can we deduce the result of a shear test a priori from a compression test? If, for example, only the shear stresses were involved in moving the dis­ locations, then this might be the case. The simplest mathematical form for such a relation is the one associated, for example, with von Mises criterion for plastic yielding; this criterion assumes that only the second invariant o f the stress tensor is involved in plastic deformation. This invariant is proportional to what is sometimes known as the octahedral shear stress, i.e. the shear stress on octa­ hedral planes in a coordinate system defined by the principal axes of stress. It is also related to that part o f the elastic strain energy due to change of shape. In the theory of glacier flow it is usually assumed that this invariant is indeed the only stress variable o f importance. To test this hypothesis one must stress ice crystals with differing amounts of longitudinal and shear stresses and see if the results can be correlated on the second invariant assumption.

Two main series of experimental results are relevant to this. Steinemann113 did tests under super­ posed shear and compressive stress. His results were analyzed by Glen33 who showed they did not fit the theory. The other tests are those o f Voytkovskiy117 who used torsion with superimposed compression, and found his results were in agreement with the second-invariant theory. For the present it seems reasonable to continue to use the second-invariant theory, while awaiting further experimental verification of its validity.

The basic idea of this theory is that the components of the strain-rate tensor are individually proportional to the components of the stress tensor, the constant of proportionality being deter­ mined by the magnitude of the second invariant of the stress tensor. The constant of proportion­ ality obviously contains within this relationship the flow law we have discussed above. A good in­ troduction to the way this theory is applied can be obtained from Paterson’s book.97

Physical processes occurring during the creep of polycrystalline ice

At the beginning of this chapter we discussed the complications that arise for a crystal with a single easily operable glide plane in deforming in a poly crystalline aggregate. There have been sev­ eral attempts to investigate experimentally just how ice manages to do this. Rigsby105 made an THE MECHANICS OF ICE 29 artificial poly crystal consisting of single crystal cubes frozen together, and deformed this in shear. In this way he was able to observe just what happened when the specimen was sheared, and he found that the grain boundaries migrated, i.e. some grains grew at the expense of others, and new crystals were formed and grew at the expense of the existing ones, i.e. recrystallization took place. He also tested randomly oriented “snow ice” and found that it too recrystallized, developing a pre­ ferred orientation in such a direction as to make shear on the basal planes easier. Wakahama122 has also investigated the processes occurring when ice is deformed; he compressed specimens while observing them in a polarizing microscope. As well as basal slip, he observed the formation of a bend plane or low-angle boundary, grain boundary slip, cavity formation at the grain boundary, and recrystallization. Using columnar ice with c-axes perpendicular to the column axes, Gold41 42 has also studied processes within the deforming ice, finding, in addition to the processes already mentioned, that cracks develop into the grains, and particularly at higher stresses continue to be formed in different places until eventually the specimen fails. He has also found evidence for non- basal glide. The recrystallization of ice during creep means that eventually the specimen will have a structure that is determined more by the recrystallization process than by the original grain size and orienta­ tion of the sample. Thus small-grained randomly oriented ice may develop larger grains with com­ plex interpenetrating shapes and with a preferred orientation. This process has been studied most carefully by Steinemann.113 The different possibilities, and in particular the effects at large stresses, have been investigated in extrusion experiments.65 It must be remembered that one result of this recrystallization is to give the specimen a preferred orientation that is related to the applied stress, so that the specimen ceases to be isotropic. The preferred orientations developed in this way are not necessarily the same as those developed by recrystallization after the deformation has ceased, or if the stress is much reduced. These phenomena will be further discussed in Chapter VI.

CHAPTER V. FRACTURE

When ice fractures.rapidly, it usually does so in a brittle manner, that is to say a crack develops somewhere in the specimen and then spreads rapidly across the whole area to produce failure. The process is thus determined either by cracks that may already be present in a specimen, or by some process which opens up cracks within the specimen. In single crystals of ice, cracks could develop from dislocation pile-ups, but such pile-ups are normally associated with work hardening, a process which ice hardly exhibits. It may, of course, be that pile-ups are so efficient at nucleating cracks that fracture takes place before work hardening has had time to become apparent. The amount of work done on the fracture of ice single crystals is not adequate to give a scientific answer to this question. In polycrystalline ice, the interactions between individual grains can cause cracks to open up in two different ways, and in this case detailed experimental work by Gold38"43 has shown these processes in action and has given data on the way cracks develop. The first kind of crack is a crack within a crystal of the aggregate. These cracks are probably formed by dislocation pile-ups, and the crack may be generated either in the crystal in which the pile-up occurs, or in a neighboring grain which is highly stressed locally by the pile-up. These cracks are normally either parallel or perpen­ dicular to the basal plane of the grain, and are approximately parallel to the compression axis in a compression test. In compression tests the appearance of such cracks does not usually lead to im­ mediate fracture of the specimen; indeed Gold found that for stresses between 6 and 10 bars, this process only occurred during primary creep. At higher stresses, however, the process continued and a reaccelerating creep was observed which led to fracture. Thus it is possible that this cracking 30 THE MECHANICS OF ICE

activity, as well as recrystallization, may be a cause for reacceleration of creep. The second kind of crack formation is development of cracks between two grains, i.e. along a grain boundary. This nor­ mally starts at a place where three or more grains meet, and is one way the crystal can adjust to the difference in plastic properties of its grains. The processes of ice fracture have been classified by Wakahama.123 All of these processes involve some kind of plastic deformation to initiate the cracks. This does not necessarily mean that they are inappropriate mechanisms for explaining an apparently completely brittle fracture, since similar mechanisms are now commonly postulated in many brittle crystalline materials. It does, however, imply that if this is the controlling mechanism for fracture, then there should be a temperature vari­ ation of the fracture strength which is connected to that for plastic deformation. As we saw in Chapter III, ice gets much harder to deform as the temperature is lowered, and there is evidence that dislocations may be incapable of movement without thermal activation. If this is so, then at temperatures below those for which dislocations can move there should be no pile-ups, no relative plastic deformation of more favorably oriented grains, and hence no small cracks developing from these causes. Under such conditions fracture would have to be initiated from cracks already pre­ sent in the ice at the beginning of the test. There is some evidence100 that if fracture tests are done fast enough, the fracture stress is inde­ pendent of temperature. However, most of the work which has been done on the fracture of ice has shown a marked increase of strength as the temperature is lowered, though not as rapid an in­ crease as the increase in resistance to plastic deformation. The classical theory of the development of a small crack to produce failure of the specimen, due to Griffith, has been modified for the case of ice by Goetze.37 He considers a material with a “microstructure” that can generate stress concentrations, such as orientation differences between grains, and then considers the conditions under which instabilities can occur. The general problem of fracture in ice, including the peculiar problems of natural lake and sea ice, is discussed in much greater detail by Weeks and Assur;127 this review should be consulted for further information on this rather confusing subject. The complications which arise due to the presence of liquid inclusions in sea ice will not be further discussed here. In the remainder of this chapter we shall consider briefly the various types of tests which have been used to study fracture of ice, and the main empirical facts which have emerged from them.

Tensile tests

Although tensile testing would seem the natural way to study fracture in ice, since fracture usually results from the presence of a tensile stress of sufficient magnitude, and since only in the tensile test is a uniform tensile stress applied to a specimen, relatively few studies of fracture have used tensile tests. The probable reason for this is the difficulty of arranging for the tensile stress to be applied without some stress concentration at the ends of the specimen, the presence of which starts future cracks at the end and so invalidates the assumption that the stress is uniform at the point of fracture. With care, however, this difficulty can be avoided,49 and when this is done ten­ sile testing is one of the most physically significant kinds of fracture testing. Results of these tests49 show that the fracture stress varies with strain-rate, but not very marl 1ly. Above strain-rates of about 10 5 s’1 the value is essentially constant. Below this rate, the material deforms markedly in creep, and the actual fracture stress is rather hard to define. The specimens used in these tests were rather bubbly, and it is not clear whether this has any marked effect on the results, but apart from this they represent the most careful investigation of tensile strength to date. THE MECHANICS OF ICE 31

Earlier tests13 have shown a rise in strength with decreasing temperature, though not so rapid a rise as with other forms of testing. A series of tests on specimens of different area and volume61 have shown that the strength S at a constant temperature (-5°C) depends on volume and area ac­ cording to the relation

S = 2.604 K"0-84 +9.2 bars where A and V are area and volume in cm2 and cm3 respectively. This kind of relation is similar to that expected theoretically on statistical grounds61 if the material is considered as being made up of a number of parallel elements, each having a definite number of imperfections. For such a theory the predicted number average of the tensile strength S is given by

S = k A l/(S V '11?+ C where k , |3and Care constants.

Ring tensile and bending (flexural) tests A common form of experimental fracture test for ice is the so-called ring tensile test,15 29 125 in which a ring of ice is compressed between parallel plates. The ring then has a very nonuniform stress distribution, and fails at the place of maximum tensile stress, whose value can be computed assuming elastic theory to apply. The reason for using this test is its ease of performance and com­ parative repeatability, but it suffers from disadvantages, among which are the fact that the region over which the stress is concentrated is comparatively small, and as we have just seen, the area is of importance in determining fracture stress. It is also true that the stress is varying rapidly at the point of maximum stress, and that other stress components besides the tensile stress are present. A detailed study of this kind of test (to be published) has shown grave difficulties in trying to cor­ relate results with tensile strengths determined in uniaxial tension. Thus the test at present should be regarded as a measure of fracture properties which can be compared with similar tests on other samples but which is not immediately usable to predict failure stresses under other conditions. The bending or flexural testing of ice has similar disadvantages to those just mentioned. It has often been performed in field conditions11 15 27 28 132 because of its simplicity in performance, but results appear inconsistent with those determined by other methods.49 127

Compression tests Like tensile tests, compression tests offer the possibility of a uniform stress condition through­ out the specimen, but this is extremely difficult to achieve in practice. Older measurements of compressive strength13’15 indicated that the strength varied with rate of loading in that an increased rate of loading produced a lower strength, but this has recently been questioned by Hawkes and Mellor49 who find that the effect disappears if more care is taken in specimen preparation and test­ ing. Their results suggest that the compressive strength data for ice fit onto the plastic flow data in a comparatively smooth curve, further evidence that the process involved in fracture is in some way connected with that for plastic flow (Fig. 15). There is no evidence in their work for any sudden ductile-brittle transition.

Shear and torsion tests Because of the difficulties associated with the tensile and compressive tests, some workers have preferred to use shear and torsion tests. In a twisted rod the stress is nonuniform, so that such a s i THE MECHANICS OF ICE

Figure 15. Limiting envelope giving maximum value o f stress)strain-rate ratio for uniaxial stress on isotropic poly crystalline ice. A t high strain-rates the data come from constant strain-rate fracture tests; at lower strain-rates they come from constant stress creep tests. (After ref. 49.)

test has similar disadvantages to the bending test, but a twisted ring is a closer approximation to a uniform shear. Torsion of a rod has been used in a number of studies13 14 and the annular shear test in two forms has also been used.100 Other forms of so-called shear test are more suspect, espe­ cially those which involve indenting a plate with a die above a hole. The stresses in these cases are not pure shear ones. The general results found with torsion and shear tests are similar to those in compression, but insufficient work has been done to say with any certainty what the effect of the different ratios of the stress components is. No triaxial tests such as have been discussed in Chap­ ter IV for creep have been published in the study of ice fracture, and this is what is required if a full understanding of the difference in behavior in shear and compression is to be obtained.

Other tests on ice fracture Very little work appears to have been done on the impact strength of ice. There is one report of measurements in a Charpy test machine.57 Testing of this kind, in which a specimen with or without a notch in it is struck and the energy absorbed in the failure measured, can throw light on the mech­ anism of failure since plastic work absorbs much more energy than simple crack propagation. It must still be remembered, however, that even a material with little energy absorption may be de­ pendent on a small amount of plastic work for initiating the crack. * There is also very little work on the fatigue testing of ice. Fatigue tests are tests in which a stress is repeatedly applied to the specimen, either in one direction or alternately in opposite direc­ tions. It is frequently found that materials can fail at comparatively low stresses under these con­ ditions. The only report of this kind of phenomenon in ice comes from work of Kartashkin72 who found that ice beams disintegrated during forced oscillations at a stress only a quarter of that needed to produce static failure. In his casexa static load of 1.5 bars with a dynamic stress in addition of about 2.5 bars was sufficient to cause failure, and a dynamic stress in addition of 2.7 bars produced the effect very quickly; the static load the ice could withstand was about 16 bars. THE MECHANICS OF ICE 33

General summary of fracture behavior The fracture stress of ice has been determined in many ways which cannot be correlated very consistently. However, three main effects can be identified. 1) The fracture stress varies with rate of stressing or rate of straining (Fig. 15). This effect is more marked in compression and shear than in tension. In careful work it appears that the stress increases in these cases, although several reports to the contrary are in the literature. 2) The fracture stress varies with temperature, increasing ap­ proximately linearly as temperature falls. 3) The fracture stress in tension depends on the volume and area of the specimen, probably because fracture depends on the statistical probability of a cer­ tain crack developing. These three effects are not fully explained, but are generally consistent with the microscopic picture of fracture as being caused by cracks which develop from difficulties in plastic flow pro­ cesses in some grains of the poly crystalline aggregate, which under appropriate circumstances act as sufficient stress raisers to allow failure cracks to run through the whole specimen. Not enough is known at present to enable us to give any general fracture law for ice; in particu­ lar the behavior of ice under triaxial testing conditions has not been studied, nor has the effect of the application of a hydrostatic pressure on the simple testing procedures. By analogy with other materials such as rocks, it is to be expected that fracture will become increasingly harder, i.e. the fracture stress will increase, as pressure rises, and observations of the magnitude of this effect are of importance in determining how geophysical ice masses can react in various circumstances. How­ ever the ratio of compressive to tensile strength is very low compared with the value found in other materials such as rocks and ceramics, and this rather surprising fact means that application of re­ sults of simple tests in ice to more complex stressing systems cannot use the relations developed for these other materials. The problem needs further investigation. The behavior of ice single crystals has been studied even less than that of poly crystals. Observa­ tions of the stresses at which creep specimens fail68 suggest that the fracture stress for single crys­ tals is also increasing as the temperature falls, and that fracture occurs brittly approximately normal to the tensile stress (and not in general on a cleavage plane of low crystallographic index), but the criterion of failure is not at all known.

CHAPTER VI. RECRYSTALLIZATION AND GRAIN GROWTH OF ICE

The phenomenon of recrystallization consists of the appearance of new grains in the material which grow at the expense of the old, pre-existing ones. It is governed by the rate at which new crystals nucleate, and also by the rate at which they grow. Physically the explanation for the phenomenon lies in the gain in free energy that can be obtained if crystals which have stored energy in them are replaced by new, strain-free crystals. Grain growth can be distinguished from recrystal­ lization by the fact that no new crystals appear; it is a phenomenon by which one grain grows at the expense of its neighbor, and the physical driving force is the decrease in surface energy which results when the average grain size increases, and the amount of grain boundary therefore decreases.

Recrystallization In ice recrystallization occurs in two different ways, which can usefully be distinguished: recrys­ tallization which takes place while iee is still forming under a stress, and recrystallization which oc­ curs after deforming stress has been removed. These phenomena also occur in other materials, though some, including many metals, do not show recrystallization while deformation is still pro­ ceeding. To understand the process, it is perhaps easiest to consider first recrystallization which 34 THE MECHANICS OF ICE occurs after a stress has been removed. In this case we can imagine the grains all to have dislocations and point defects in them as a result of the plastic deformation, and that these give an increase in internal energy which is available to drive a recrystallization. The question of how a new nucleus in a rather different crystallographic orientation appears is not one that can easily be answered; it probably arises from local rotations occurring in the plastic deformation. The difference in orienta­ tion is necessary for there to be a disordered grain boundary across which material can move and join to the new grain. If there were no such orientation difference, or if it were small enough so that the boundary were essentially a wall of dislocations, it is hard to see how the stored energy, particularly that in the form of dislocations, could be released. If, however, a new nucleus appears, it will grow at the expense of the old grains, the rate being determined by the amount of energy being released. This will continue until new grains have filled the whole volume. The final grain size will therefore be determined by the nucléation rate and the growth rate. If, as seems to be the case in ice,31 113 the grain size is smaller the larger the stored strain energy, then we must assume that the nucléation rate increases more rapidly with stored energy than the growth rate. This is not a very surprising result; it is true for many other materials, and it is also what is to be expected if nucléation requires very large local disturbances of the lattice, which become much more common when the strain is very high. In materials which do not recrystallize under stress which is still active, it is common to describe the recrystallization in terms of the strain the material has undergone, and this is clearly a useful measure of the strain energy stored. When we turn to consider ice which is recrystallizing while creep is proceeding, however, we must recognize that the process is much more complicated. It is probably correct to discuss the strain the specimen must have before any recrystallization begins, but after this has happened, the newly grown strain-free grains will be part of the deforming aggre­ gate and will begin themselves to acquire strain. They too will be available for recrystallization when they have acquired enough strain, and so, as time proceeds, the ice can be expected to recrys­ tallize repeatedly. In these circumstances the total strain of the specimen is less important, and the important parameter is the stress which is being applied to the sample. These processes have been studied in ice most exhaustively by Steinemann.113 He deformed thin ice samples and watched the progress of recrystallization under polarized light. He was able to show that the general picture described above is true for ice, though of course his thin samples are not completely like a three-dimensional mass of ice crystals. Glen31 also studied these phenomena. In his case the observations were made on the actual samples he used for polycrystalline creep tests. Thus the structures he observed were those formed by the three-dimensional situation, but of course he could not observe them develop in detail. He did find that the resulting grain size was smaller the larger the stress on the crystal. This result also follows from the fact that nucléation rate is a more rapid function of strain than growth rate, though the connection is not quite so obvious; it arises because at a higher stress the average strain in the crystals must be higher, and hence the re­ crystallization is similar to that for a higher stored energy or higher strain. The effect is quite marked, and is recognized as being of importance in determining the grain size in glaciers, where the grain size is found to decrease in places of high stress, such as beneath ice falls. A further question which must be asked is whether during this kind of recrystallization the con­ ditions will eventually become steady, with recrystallization occurring all the time, or whether it will be cyclic.65 113 The answer would appear to be that at stresses below about 7 bars at -4.8 C the phenomenon is periodic, while above this stress it is continuous. Steinemann attributes the difference to the ability of the ice to reach the necessary minimum strain for recrystallization in the new grains during the time it takes for the recrystallization process to be completed. After a process of recrystallization with the stress on has finished, a further process of recrystal­ lization with the stress removed may well take place, and will in general produce a quite different, THE MECHANICS OF ICE 35 larger grain size. If the stress is suddenly changed in the middle of a test, a similar kind of phenom­ enon might take place. It is for this reason that these two kinds of jecrystallization must only be regarded as special cases of a general phenomenon of recrystallization under changing stress condi­ tions. The general situation, which is much nearer to what happens in geophysical situations, is however too complicated to have been studied properly in the laboratory. The shape of the new grains is of importance, as is also their orientation. Even if the old grains were equiaxed and randomly oriented, the new ones often are not.71 113 The recrystallized grains frequently have irregular, branching shapes, and have a preferred orientation related to the stress under which they are formed. In shear tests this preferred orientation seems to have a single max­ imum perpendicular to the plane of shear,113 sometimes accompanied by a second maximum ro - tated away from the first in the direction of the rotation accompanying the simple shear. Com­ pression experiments produce a ring of maximum c-axis orientations around the direction of com­ pression.71 113 In both cases this recrystallization is producing a preferred orientation likely to be more easy to deform than the pattern from which it came. The textures found are quite different from those found in glacier ice under what appears to be similar stress conditions.71 Kamb attri­ butes this difference to the vastly different time scales in the two cases, but no convincing physical explanation has been suggested. The recrystallization of ice in compression tests has also been observed by Mellor and Testa,88 who found that grains of a quite different size developed in conical regions beneath each end of the specimen. This experiment, white not telling us very much about the recrystallization process, shows how recrystallization can be used as evidence that the strain was nonuniform in a test which is often assumed, unjustifiably, to be more uniform than most others. Recrystallization in thin specimens of ice has klso been observed using a polarized light technique by Wakahama,122 who finds that the effect does not occur in a plate containing only a few grains. This is further evidence that the restraints which the different grains in a polycrystal exert on each other are of great importance in building up the stored strain energy needed for recrystallization. A similar result follows from Rigsby’s experiment using an artificial polycrystal.105 In his case a specimen originally consisting of 16 cubes of ice in different orientations frozen together to make a square “polycrystal” was deformed in a shearing frame for about 2 months, after which time it had no fewer than 135 different grains. In another experiment, Rigsby105 started with randomly oriented “snow ice” and, after repeated shearing in the shear frame, a pattern of orientations was found which had two strong maxima in the two difections (at 90°) of maximum shear in the speci­ men. Again this is a recrystallization tending to give crystals more easy to deform than the original random distribution.

Grain growth in ice Grain growth in ice in the absence of a stress has also been investigated. The most careful experi­ ments are those of Jellinek and Gouda;63 somewhat similar tests have been reported by Roos.107 In both cases thin samples were used to assist in seeing the phenomenon; it can be expected that, as a result, the growth will be somewhat less rapid than would occur in a full, three-dimensional situation. The grain size of the specimens increased according to the relationship

b = K tn where b is the average grain diameter and t the time. For pure ice the value of n was about 0.30, and K varied from 4.85x 10“2 cm day“” at -3°C to 1.12x 10"2 cm day“” at -36°C. Tests on ice 36 THE MECHANICS OF ICE doped with NaCl were somewhat similar, but the temperature variation was greater - the pure ice grew more slowly than doped ice above -10.5°C and faster below. From these data an activation energy can be deduced; it is 0.24 eV for pure ice and 0.31 eV for the doped ice. These are very low activation energies, much lower than the activation energy for self diffusipn. It so happens that the pure ice activation energy is similar to that for the movement of a Bjerrum defect, but this is prob­ ably accidental. It seems likely that the process involved is essentially one at the grain boundary it­ self, and may involve the movement of more than one water molecule at a time. In somewhat less quantitative experiments, Roos107 found the rate of change of average diameter to be approximately inversely proportional to the annealing time; he also found the boundary angles tended to 120° irrespective of the number of sides per grain, that both grain shape and relative ori­ entation affected boundary migration rates, and that after long times there was a strong tendency for the surviving grains to have their c-axes parallel to the plane of the ice surface. This last obser­ vation shows that the surface was indeed exerting a large effect on the grain growth process. In a truly randomly oriented sample with no applied stresses or large, flat surfaces, there is no reason for a preferred orientation to develop as a result of grain growth. If a stress is present, then there may be such a tendency even without plastic deformation and the corresponding recrystalliza­ tion phenomenon. Kamb69 has considered theoretically the possibility of preferential growth of those grains which are favored because of the elastic anisotropy, and has attempted to explain some recrystallization textures in this way. Ice crystals resulting from long-term grain growth in relatively stagnant glacier tongues are among the most perfect crystals known. Those collected from the Mendenhall Glacier in Alaska have been used in numerous experimental investigations.

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62. Jellinek, H.H.G. (1960) Plastic deformation of thick-walled snow-ice cylinders under hydrostatic pressure. USA SIPRE Research Report 63. AD 696401. 63. Jellinek, H.H.G. and V.K. Gouda (1969) Grain growth in poly crystalline ice. Physica Status Solidi, vol. 31, no. 1, p. 413-423. 64. Jona, F. and P. Scherrer (1952) Die elastischen Konstanten von Eis-Einkristallen. Helvetica Phys- ica Aeta, vol. 25, fasc. 1-2, p. 35-54. 65. Jonas, J.J. and F. Müller (1969) Deformation of ice under high internal shear stress. Canadian Journal o f Earth Sciences, vol. 6, no. 4, pt. 2, p. 963-968. 66. Jones, S.J. and J.W. Glen (1968) The mechanical properties of single crystals of ice at low tempera­ tures. Union de Géodésie et Géophysique Internationale, Association Internationale d’Hydrol- ogie Scientifique, Assemblée Générale de Berne, 1967. (Commission de Neiges et Glaces.) Rapports et discussions, p. 326-340. 67. Jones, S.J. and J.W. Glen (1969) The effect of dissolved impurities on the mechanical properties of ice crystals. Philosophical Magazine, Eighth Ser., vol. 19, no. 157, p. 13-24. 68. Jones, S.J. and J.W.. Glen (1969) The mechanical properties of single crystals of pure ice. Journal o f Glaciology, vol. 8, no. 54, p. 463-473. 69. Kamb, W.B. (1959) Theory of preferred crystal orientation developed by crystallization under stress. Journal o f Geology, vol. 67, no. 2, p. 15 3-170. 70. Kamb, W.B. (1961) The glide direction in ice. Journal o f Glaciology, vol. 3, no. 30, p. 1097-1106. 71. Kamb, W.B. (1964) Glacier geophysics. Science, vol. 146, no. 3642, p. 353-365. 72. Kartashkin, B.D. (1947) Work referred to in Voytkovskiy, K.F. (1960) Mekhanicheskiye svoystva l’da (Mechanical properties of ice). Izd. Akademii Nauk SSSR, Moscow. English translation Air Force Cambridge Research Center, AFCRL-62-838; American Meteorological Society AMS-T-R-391+. 73. Khrushchov, M.M. and Ye.S. Berkovich (1960) Izucheniye tverdosti l’da (A study of the hardness of ice). Moscow, Izdatel’stvo Akademii Nauk SSSR’. 74. Kneser, H.O., S. Magun and G. Ziegler (1955) Mechanischen Relaxation von einkristallinen Eis. Naturwissenschaften, 42, Jahrg., Ht. 15, p. 437. 75. Krausz, A.S. (1963) The creep of ice in bending. Canadian Journal o f Physics, vol. 41, no. 1, p. 167-177. 76. Krausz, A.S. (1968) An experimental investigation of strain relaxation in ice. Scripta Metal- lurgica, vol. 2, no. 11, p. 615-619. 77. Krausz, A.S. (1968) A rate theory of dislocation mobility. Acta Metallurgical0I. I 6, no. 7 , p. 897-902. 78. Krausz, A.S. (1969) A rate theory of strain relaxation. Materials Science and Engineering, vol. 4, no. 4, p. 193-197. 79. Kuroiwa, D. and K. Yamaji (1959) Internal friction of poly crystalline and single-crystal ice. Low Temperature Science, Ser. A, no. 18, p. 97-114. 80. Kuroiwa, D. (1964) Internal friction of ice. Contributions from the Institute o f Low Tempera­ ture Science, Ser. A, no. 18, 62 p. 81. Larsson, K.E. and U. Dahlborg (1962) Some vibrational properties of solid and liquid H20 and D20 derived from differential neutron cross-section measurements. Journal of Nuclear Energy, Parts A and B, vol. 16, no. 2, p. 81-89. 82. Levi, L., E.M. de Achaval and E. Suraski (1965) Experimental study of non-basal dislocations in ice crystAis. Journal of Glaciology, vol. 5, no. 41, p. 691-699. 83. Mae, S. (1968) Void formation during non-basal glide in ice single crystals under tension. Philosophical Magazine, Eighth Ser., vol. 18, no. 151, p. 101-114. 84. McConnel, J.C. (1891) On the plasticity of an ice crystal. Proceedings o f the Royal Society o f London, vol. 49, no. 299, p. 323-343. 85. Mellor, M. (1964) Properties of snow. USA CRREL CRSE Monograph III-Al. AD 611023. 86. Mellor, M. and J.H. Smith (1967) Creep of snow and ice. In Physics o f Snow and Ice: Interna­ tional Conference, 14-19 August 1966, Sapporo, Japan, vol. 1, pt. 2, p. 843-855. 87. Mellor, M. and R. Testa (1969) Creep of ice under low stress. Journal o f Glaciology, vol. 8, no. 52, p. 147-152. 88. Mellor, M. and R. Testa (1969) Effect of temperature on the creep of ice. Journal o f Glaciology, vol. 8, no. 52, p. 131-145. 89. Mügge, O. (1895) Ueber die Plasticitât der Eiskrystalle. Neues Jahrbuch fur Minéralogie, Géolo­ gie und Palaeontologie, II Bd., p. 211 -228. 40 THE MECHANICS OF ICE

90. Muguruma, J.and A. Higashi (1963) Non-basal glide bands in ice crystals. Nature, vol. 198, no. 4880, p. 573. 91. Muguruma, J., S. Mae and A. Higashi (1966) Void formation by non-basal glide in ice single crystals. Philosophical Magazine, Eighth Ser., vol. 13, no. 123, p. 626-629. 92. Muguruma, I. (1969) Effects of surface condition on the mechanical properties of ice crystals. Journal of Physics D (British Journal o f Applied Physics), Ser. 2, vol. 2, no. 11, p. 1517- 1525. 93. Nakahara, Y. ( ) Phonon spectrum and thermal neutron spectrum in light water ice. Journal o f Nuclear Science and Technology, vol. 5, no. 12, p. 635-642. 94. Nakamura, T. and S.J. Jones (1970) Softening effect of dissolved hydrogen chloride in ice crystals. Scripta Metallurgica, vol..4, no. 2, p. 123-126. 95. Nakaya, U. (1958) Mechanical properties of single crystals of ice. Part 1: Geometry of deforma­ tion. USA SIPRE Research Report 28. AD 216992. 96. Nye, J.F. (1957) Physical properties o f crystals: their representation by tensors and matrices. Oxford: Clarendon Press. 97. Paterson, W.S.B. (1969) The physics o f glaciers. Oxford: Pergamon Press. 98. Prask, H., H. Boutin and S. Yip (1968) The frequency distribution of ice by neutron scattering. Developments in Applied Spectroscopy, vol. 6, p. 265-276. 99. Proctor, T.M. Jr. (1966) Low-temperature speed of sound in single-crystal ice. Journal o f the . Acoustical Society o f America, vol. 39,tto. 5, pi* 1, p. 972-977. 100. Raraty, L.E. and D. Tabor (1958) The adhesion and strength properties of ice. Proceedings o f the Royal Society, Ser. A, vol. 245, no. 1241, p. 184-201. 101. Readey, D.W. and W.D. Kingery (1964) Plastic deformation of single crystal ice. Acta Metal- lurgica, vol. 12, no. 2, p. 171-178. 102. Readings, C.J. and J.T. Bartlett (1968) Slip in single crystals of ice. Journal of Glaciology, vol. 7, no. 51, p. 479-491. 103. Renker, B. (1969) Phonon dispersion in D20-ice. Physics Letters A , vol. 30, no. 9, p. 493-494. 104. Rigsby, G.P. (1957) Effect of hydrostatic pressure on velocity of shear deformation of single crystals of ice. USA SIPRE Research Report 32. AD 143290. Also Journal o f Glaci­ ology, vol. 3, no. 24, p. 273-278, 1958. 105. Rigsby, G.P. (1960) Crystal orientation in glacier and in experimentally deformed ice. Journal o f Glaciology, vol. 3, no. 27, p. 589-606. / 106. Roethlisberger, H. (1966) Ultrasonic pulse measurements in anisotropic lake ice. USA CRREL Research Report 126. AD 642110. 107. Roos, D.v.d.S. (1966) Two-dimensional grain growth in ice. Journal o f Glaciology, vol. 6, no. 45, p. 411-420. 108. Rüter, H. (1968) Berechnung der Richtungsabhangigkeit der P-Wellen-Geschwindigkeit in poly- kristallinen Eisproben mit vorgegebener C-Achsen-Verteilung aus den elastischen Konstanten des Einkristalls. Polarforschung, Bd. 6, Jahrg. 37, Ht. 1-2, p. 175-177. 109. Schiller, P. (1958) Die mechanische Relation (sic] in reiner Eiseinkristallen. Zeitschrift für Physik, Bd. 153, l.H t.,p . 1-15. 110. Schulz, H. (1961) Die mechanische Relaxation in Eis-HF-Mischkristallen verschiedener HE Konzentration. Naturwissenschaften, 48, Jahrg, Ht. 22, p. 691. 111. Shawyer, R.E. and P. Dean ( ) Atomic vibrations in hexagonal ice I. Discussions of the Fara­ day Society, ho. 48. 112. Steinemann, S. (1954) Results of preliminary experiments on the plasticity of ice crystals. Jour­ nal o f Glaciology, vol. 2, no. 16, p. 404-412. See also Beitrage zur Hydrologie der Schweis, Hydrologie, No. 10, 1958. 113. Steinemann, S. (1958) Experimentelle Untersuchungen zur Plastizität von Eis. Beitrage zur Geologie der Schweiz, Geotechnische Serie; HyMrbiogy, no. 10, 72 p. 114. Tammann, G. and N. Salge (1928) Der Einfluss des Druckes auf die Reibung beim Gleiten längs der Gleitebenen von Kristallen. Neues Jahrbuch für Mineralogie, Geologie und Paläontologie, Beilage Bd. 57, Abt. A, Erste Hälfte, p. 117-130. 115. Tegart, W.J.Mc.G. (1964) Non-basal slip as a major deformation process in the creep of polycrys­ talline ice. Journal o f Glaciology, vol. 5, no. 38, p. 251-254. 116. Trevino, S., H. Prask, T. Wall and S. Yip (|9é8) Lattice dynamics of hydrogen-bonded crystals. In Neutron inelastic scattering Proceedings of a Symposium held by the International Atomic Energy Agency in Copenhagen 20-25 May 1968, Vienna, LA.ILA., vol. 1, p. 345-365. THE MECHANICS OF ICE 41

11?. Voytkovskiy, K.F. (1960) Mekhanicheskiye svoystva I’da (The mechanical properties of ice). Moscow, Izd. Akademii Nauk. English translation AFCRL-62-838, AMS-T-R391+, 1962. 118. Voytkovskiy, K.F. (1967) The relaxation stresses in ice. In Physics o f Snow and Ice: International Conference, 14-19 August 1966, Sapporo, Japan (H. Oura, Ed.), vol. 1, pt. 1, p. 329-337. 119. Vyalov, S.S. (1967) On theory of the flow of glaciers. In Physics o f Snow and Ice: International Conference, 14-19 August 1966, Sapporo, Japan (H. Oura, Ed.), vol. 1, pt. 1, p. 349-356. 120. Vyalov, S.S. and V.F. Yermakov (1967) Simplified method of testing ice for creep and relaxation. In Physics o f Snow and Ice: International Conference, 14-19 A ugust 1966, Sapporo, Japan (H. Oura, Ed.), vol. 1, pt. 1, p. 339-347. 121. Wakahama, G. (1962) On the plastic deformation of ice. Low Temperature Science, Ser. A, vol. 20, p. 57-100. 122. Wakahama, G. (1964) On the plastic deformation of ice. Plastic deformation of polycrystalline ice. Low Temperature Science, Ser. A, vol. 22, p. 1-24. 123. Wakahama, G. (1965) Internal fracture of ice. Low Temperature Science, Ser. A, vol. 23, p. 39-50. 124. Walz, E. and S. Magun (1959) Die mechanische Relaxation in Eis-NH4 F-Mischkristallen. Zeitschrift fur Physik, Bd. 157, 2. Ht., p. 266-274. 125. Weeks, W.F. (1962) Tensile strength of NaCl ice. Journal o f Glaciology, vol. 4, no. 31, p. 25-52. 126. Weeks, W.F. and A. Assur (1967) The mechanical properties of sea ice. USA CRREL CRSE Monograph II-C3. AD 662716 127. Weeks, W.F. and A. Assur (1969) Fracture of lake and sea ice. USA CRREL Research Report 269, AD 69=7750. 128. Weertman, J. (1963) The Eshelby-Schoeck viscous dislocation damping mechanism applied to the steady-state creep of ice. In Ice and snow: Properties, processes and applications (W.D. Kingery, Ed.). Cambridge, Mass.: M.I.T. Press, p. 28-33. 129. Weertman, J. (1968) Dislocation climb theory of steady-state creep. Transactions o f American Society for Metals, vol. 61, no. 4, p. 681-694. 130. Weertman, J. (1969) The stress dependence of the secondary creep rate at low stresses. Journal o f Glaciology, vol. 8, no. 54, p. 494-495. (Letter. See also reply by M. Mellor, ibid, p. 495.) 131. Whitworth, R.W. ( ) Charged dislocations in ionic crystals. Advances in Physics. 132. Wilson, J.T. and J.M. Höreth (1948) Bending and shear tests on lake ice. Transactions American Geophysical Union, vol. 29, no. 6, p. 909-912. 133. Woerner, S. and S. Magun (1959} Mechanische Dämpfung von D^O-Einkristallen. Naturwissen­ schaften, 46. Jahrg., Ht. 17, p. 509-510. 134. Young, LG. and R.E. Salomon (1968) Dielectric behavior of ice with HCl impurity. Journal o f Chemical Physics, vol. 48, no. 4, p. 1635-1644.

43

APPENDIX: CRYSTALLOGRAPHIC INDICES USED TO SPECIFY PLANES AND DIRECTIONS IN ICE

Since ice Ih, the usual form of ice, is hexagonal, the usual system for labeling planes and directions in ice is one that involves four numbers. Since this differs from the more usual system of Miller indices involving three numbers used in, for example, cubic crystals, it may be helpful to readers of this Monograph to explain how this system is set up. This appendix does not aim to give a full treatment of Miller-Bravais indices (as these four-number indices are called); for this the reader is referred to any good book on crystallography, such as Phillipps, F.C. (1946) Introduc­ tion to crystallography. London and New York: Longmans. The reason for using four-number indices is that, in the hexagonal system, there are three di­ rections in the basal plane that are crystallographically equivalent. If two of these are taken as the a and b axes of the crystal, and if the c axis is taken, as usual, perpendicular to the basal plane, then these three crystallographically equivalent directions have Miller indices (100], [010], and [TTo]. Thus the fact that they are crystallographically equivalent is hidden, whereas by use of Miller-Bravais indices it becomes clear. It is perhaps easiest to introduce Miller-Bravais indices for planes first. In this case the rule for finding the four numbers remains exactly the same as the rule for finding the three numbers in conventional Miller indices. If a plane intersects the crystallographic axes at points which are at distances a/h, b/k, c/l, where a, b and c are the lengths of the sides of the unit cell, then the plane is designated (h k l). For the hexagonal crystal, we simply insert a fourth axis, so that in the basal plane we have a, b, d, all of equal length and at 120° to each other as well as c, still at right angles to all the others and of different length. A plane now intersects these axes at dis­ tances a/ti, b/k, d/i, c/l, and (hki 2) are the indices of the plane. It follows from geometry that h + k + i = 0. With these indices a plane perpendicular to the a axis and half a unit cell away from the ori­ gin has indices (2110), and[crystallographically similar planes perpendicular to the b and d axes have indices (1210) and (1120), so that their symmetry is now obvious. When we come to give indices to directions, the process is not quite so simple. The usual rule is to make up the direction by adding multiples of unit vectors defined by the sides of the unit cell; thus the [h kl] direction is in the direction 2ia + kb + 1c where a, b and c are the unit vectors. If we proceed to add a fourth unit vector d, and define our direction as ha + kb + id + 2c, then any direction can be produced an infinite number of ways, since a + b + d = 0. We therefore impose a further condition that the sum of the first three indices, h + k + i, must equal zero, and to do this we add or subtract equal amounts to each of the three indices until the condition is fulfilled. In this way we here insist on this as an additional condition, whereas in the case of planes the same result happened automatically. To see how this works, let us find out the Miller-Bravais indices for a direction parallel to the a axis. Our first thought is that this should be [1000]. However this is not in accord with the rule that h + k + 2 = 0, so we have to subtract n from each until this is true. Since in our case h + k + 2 = 1, we must subtract % from each so as to subtract 1 in all; we thus get % % %• However it is a general rule that we do not have fractions in crystal indices, so we multiply by 3 to get rid of them, and finally arrive at the indices [2110] for the direction. Directions parallel to the b and d axes will similarly have indices [1210] and [1120]. We thus see that the symmetry is again apparent, and furthermore that a direction with a given index is (as in cubic crystals) perpendicular to a plane of the same index. Readers are warned, however, that this is only true for planes and directions with a zero in the final place. Nevertheless, for these rather common cases, it is a very useful result. Of course, it is also true that (0001), the basal plane, is perpendicular to [0001], the hexad axis direction.