Hooke R.L. Principles of glacier mechanics

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Summary 41

shrinking. The uncertainties for smaller glaciers and ice caps are lower,

however, and suggest that melting of these ice masses may be responsible

for 15% to 25% of the sea-level rise. (In fact, Arendt et al.(2002) esti-

mate that in the late 1990s the mass loss from Alaskan glaciers alone was

−96 ± 35 km

−1

,which is more than the estimate in Table 3.2 for all

valley glaciers and ice caps.) Other contributions to sea level are thermal

expansion of the oceans (0.5 ±0.2 mm a

−1

), melting of permafrost (0.025

±0.025 mm a

−1

), sedimentation in the oceans (0.025 ±0.025 mm a

−1

and terrestrial storage in lakes and groundwater reservoirs (−0.35 ±

0.75 mm a

−1

Although the mass balance data for Greenland are ambiguous, yield-

ing a net balance of −44 ±53 Gt a

−1

, the pattern is suggestive. The obser-

vational data show that the ice sheet is growing thicker in the interior, at

least locally, as warmer air transports more moisture inland. However, it

is thinning along the margins where the increased temperature results in

more melting (Krabill et al., 2000; Thomas et al., 2000). As mentioned

earlier, this is precisely the pattern inferred from Greenland ice cores for

the end of the Younger Dryas (Alley et al., 1993).

Summary

In this chapter we discussed snow accumulation and the transformation

of snow to ice. We found that in polar environments where there is little

if any melting, the physical and chemical stratigraphy in an annual layer

of snow persists for many thousands of years and can be used to date

the ice.

We then deﬁned some terms used to discuss mass balance, particu-

larly summer, winter, and net balance, and used a perturbation approach

to study the inﬂuence of winter balance, temperature, and radiation on

net balance. By comparing observed variations in these parameters with

calculated values, it became clear that the net balance of glaciers in con-

tinental environments was sensitive, primarily, to summer temperature,

while that of glaciers in maritime areas was sensitive to both winter bal-

ance and summer temperature. Radiation balance, principally resulting

from differences in cloud cover, could play a role in either environment.

The lower budget gradient, and consequently the more sluggish behavior

of polar glaciers compared with their temperate counterparts, turned out

to be largely related to the shorter melt season in polar environments.

On ice sheets, we also noted that accumulation decreases with distance

from the moisture source.

We then discussed the importance of calving and bottom melting in

mass balance, and discovered that on tidewater glaciers calving can lead

to retreats that are, at best, only weakly related to climate. On ice sheets,

calving turns out to be a dominant process of mass loss. Bottom melting

42 Mass balance

is an important component of mass balance on some fast-moving valley

glaciers and beneath ice shelves.

Finally, we found that variations in intermediate and large scale

weather patterns that we are just beginning to understand can result

in asynchronous mass balance patterns on glaciers only a few hundred

kilometers apart.

Chapter 4

Flow and fracture of a crystalline

material

Before proceeding to a more theoretical discussion of the dynamics of

glaciers, it will be useful to present a brief introduction to the voluminous

literature on deformation or creep of ice. We will begin by looking at

deformation processes on an atomic scale and then introduce empirical

and semi-empirical relations that provide a macroscopic description of

the deformation. Finally, we will show how principles of linear fracture

mechanics can be used to predict crevasse depth.

Crystal structure of ice

There are nine known crystalline forms of ice, but seven of them are

stable only at pressures in excess of about 200 MPa, and the eighth, a

cubic form, ice Ic, is stable only at temperatures below about −100

◦

(Figure 4.1). As the highest pressures and lowest temperatures in glaciers

on Earth are about 40 MPa and −60

◦

C, respectively, these eight forms

need not concern us. We thus restrict our attention to the common form

of terrestrial ice, ice Ih.

The structure of ice Ih is shown, in stereoscopic view, in Figure 4.2a.

It is a hexagonal mineral (hence the “h”) with a rather open structure in

which every oxygen atom, represented by the large circles in Figure 4.2a,

is bonded to four additional oxygen atoms at the corners of a tetrahedron.

The tetrahedra are joined together in such a way that the oxygens form

hexagonal rings with the O=O bonds zigzagging lightly up and down as

one progresses around the ring (Figure 4.2b); three of the oxygens thus

lie 0.09 nm above the other three. The plane of these rings is called the

basal plane of the crystal structure.

44 Flow and fracture of a crystalline material

Liquid

Pressure, MPa

Temperature,

2000

1500

2500

1000

500

−100

−50

50 100

VIII

VII

III

Figure 4.1. Part of the

phase diagram of water. (After

Kamb, 1965, Figure 1.) The

various polymorphs of ice

are designated by Roman

numerals. Ice IV is a

metastable phase, unstable

everywhere with respect to

ice V. Ice Ic is also metastable

with respect to ice Ih.

The fourth oxygen atom in the tetrahedron is ∼0.28 nm above or

below that in the center of the tetrahedron. A line parallel to this bond,

and hence normal to the basal plane is called the c-axis.

It is evident from Figure 4.2b that there are many more O=O bonds

within a basal plane than there are between basal planes. Thus, bonding

between basal planes is much weaker than that within the basal plane.

Around each oxygen atom there are, of course, two hydrogen atoms.

The hydrogen atoms, represented by the small circles in Figure 4.2a, lie

on the bonds between the oxygen atoms. They are situated close to the

oxygen to which they are bonded. As each oxygen atom is bonded to

four other oxygens, only two of these hydrogen sites, selected at random,

are occupied.

There are two hydrogen sites along each O=O bond. Normally only

one of these is occupied. Situations in which neither site is occupied are

called Bjerrum L defects, and situations in which both sites are occupied

are called Bjerrum D defects,orjust L and D defects, respectively.

Dislocations

Another type of defect in a crystal is the dislocation. Dislocations are

places where the crystal structure is discontinuous or offset in some

way. The two basic types of dislocation, the edge dislocation and the

Dislocations 45

0.09 nm

0.28

(a)

(b)

-axis

Figure 4.2. (a) Stereographic view of the structure of ice Ih, viewed down the

c-axis. Only half of the possible hydrogen sites, indicated by small circles, are

occupied. (After Hamilton and Ibers, 1968.) (b) Structure of ice Ih viewed normal

to the c-axis. Two of the hexagonal rings are shown. Short lines leading upward

and downward from these rings are bonds to rings above and below. The

oxygen shown with an open circle is the center of a tetrahedron, part of which is

shown by the light dashed lines. (Modiﬁed from Hobbs, 1974, Figure 1.7.)

screw dislocation, are illustrated in Figure 4.3.Virtually all crystalline

materials contain dislocations.

Dislocations play a vital role in the deformation or creep of crys-

talline materials. If one tried to deform the perfect crystal in Figure 4.3a

by shearing the top three layers of atoms over the bottom two, the stress

46 Flow and fracture of a crystalline material

(a)

(b)

(c)

Figure 4.3. (a) A perfect crystal. (b) An edge dislocation. (c) A screw

dislocation. (Modiﬁed from Hull, 1969,p.17.)

required would be enormous as every one of the bonds indicated by an

“×”would have to be broken simultaneously.Incontrast, the crystal in

Figure 4.3b would deform much more easily because the bonds could be

broken sequentially, one at a time. For instance, the bond between E and

F could be broken and a new bond formed between D and F. Calculations

show that, in the absence of dislocations, crystalline materials could not

possibly deform under the stresses at which they are observed to deform.

In fact, it was through such theoretical studies that the existence of dis-

locations was ﬁrst inferred.

Upon application of a stress, the number of dislocations in a crys-

tal increases. Some of these new dislocations are generated at Frank–

Reed sources.AFrank–Reed source consists of a dislocation lying

between two points at which the dislocation is ﬁxed, called pinning points

(Figure 4.4). Impurities or immobile tangles of dislocations may serve

as pinning points. When a stress is applied, this dislocation is bowed

Dislocations 47

"a"

(a)

(b)

Figure 4.4. Generation of

dislocations at a Frank–

Reed source. Each line in

(a) represents a successive

position of a dislocation as it

is bowed out between two

pinning points. (b) shows the

ﬁnal stage with the new

dislocation expanding

outward and another

dislocation between the

pinning points.

Grain-boundary

slip

Dislocations

Figure 4.5. Generation of

dislocations at a three-grain

intersection due to

grain-boundary slip.

out until it meets itself at “a”. At this point, dislocations coming from

opposite directions are of opposite sign, and the dislocation is locally

annihilated, leaving a ring and a new dislocation between the pinning

points (Figure 4.4b). This new dislocation can then repeat the process,

so there is a continuous source of dislocations. The dislocation is of

the edge type ahead of and behind the source, of the screw type at the

sides, and of mixed type at intermediate positions. Some dislocations

generated by a Frank–Reed source never complete a full cycle but rather

multiply by spreading to neighboring planes, a process called multiple

cross glide (Hull, 1969, pp. 165–7).

Dislocations also form at points of stress concentration on grain

boundaries. For example, shear along a discrete atomic plane in one

crystal can result in an offset of the crystal boundary. To accommodate

this offset, the neighboring crystal must also yield, so dislocations are

formed at the boundary and move into this crystal.

Slip along grain boundaries is believed to occur at temperatures

above −10

◦

C. This, too, results in stress concentrations in neighbor-

ing crystals, and hence in generation of dislocations in these crystals

(Figure 4.5).

48 Flow and fracture of a crystalline material

Kink

Direction of movement of kink

Direction of movement

of dislocation

- -

+ +

+ + +

Figure 4.6. Lateral movement of kinks causing forward movement of a

dislocation. The symbol

represents situations in which a normal bond would

form by passage of a kink. The symbol

represents situations in which a

Bjerrum defect would be formed.

Dislocations move by formation of kink pairs (Figure 4.6), followed

by lateral movement of the kinks. As long as the bond formed by move-

ment of a kink is a normal bond, with only one hydrogen between two

oxygens, the kink can move readily. However, if the movement would

result in a Bjerrum defect the energy required for movement would be

much higher. It is presumed that in such situations, movement of the kink

is delayed until diffusion or rearrangement of the hydrogen atoms (pro-

ton rearrangement) results in a geometry such that the kink can migrate

without formation of a Bjerrum defect. As shown in Figure 4.6, there

may be a number of kinks along a dislocation line. Two kinks moving

toward one another will annihilate each other when they meet, resulting

in advance of the dislocation line.

Rate-limiting processes

The rate of deformation of a crystal or of a polycrystalline aggre-

gate depends on how rapidly dislocations can move. This, in turn, may

depend upon factors such as the effectiveness of the mechanisms resist-

ing motion, the ability of a dislocation to move from one atomic plane

to another, and the orientation of the atomic plane in which the dislo-

cation is moving. Usually, one process is signiﬁcantly more important

than the others, principally because it is more effective than the others in

retarding dislocation motion. This process is called the rate-controlling

or rate-limiting process.

It is, in general, not easy to identify the rate-limiting process. This is

because it is different in different materials, and within any one material it

changes with temperature and stress, and possibly also impurity content.

Furthermore, the rate-limiting process may be different in single crystals

and in polycrystalline aggregates.

Rate-limiting processes 49

In the next few paragraphs, we describe some possible rate-limiting

processes in ice and present evidence for them.

Drag as the rate-limiting process

At the moderate stress levels normally found in glaciers, dislocations

moving in a crystallographic plane are restrained in their motion by a

number of drag mechanisms. The velocity of such a dislocation is given

by:

v = cσ b

−

Rθ

(4.1)

where c is a constant of proportionality that depends, in part, on

Boltzman’s constant; σ is the applied stress; b is the Burgers vector

of the dislocations (a measure of the distortion of a crystal produced

by a dislocation, and approximately equal to the atomic spacing in the

crystal (Hull, 1969, pp. 19)); R is the gas constant; θ

is the temperature

in Kelvins; and Q is the activation energy for the rate-limiting defor-

mation mechanism (Weertman, 1983). Derivation of this equation, as

well as some others in this chapter, is beyond the scope of this book.

Noteworthy, however, is the dependence on very fundamental physical

parameters.

A brief comment on activation energy is in order. The activation

energy is the magnitude of an energy barrier that must be overcome

for a kinematic process to occur. Each kinematic process has its own

activation energy, so there is an activation energy for self-diffusion of

hydrogen and oxygen in ice (≈60 kJ mol

−1

), an activation energy for

creep of ice (≈79 kJ mol

−1

), and so forth.

The creep rate, ˙ε, resulting from movement of dislocations in a crys-

tallographic plane is:

˙ε = αbρ

v (4.2)

(Weertman, 1983), where α is a constant with dimensions of length and

is the dislocation density (m

−2

); α depends on the orientation of the

slip planes, but is ∼4.5 × 10

−10

m. Of interest is the fact that the steady-

state dislocation density, which reﬂects a balance between the applied

stress and the internal stress from dislocations, is given by:

≈



µb



(4.3)

(Weertman, 1983; Alley, 1992), where µ is the shear modulus. (The table

on p. xiv gives deﬁnitions of parameters such as µ and values appropriate

for ice.) Thus, combining Equations (4.1), (4.2), and (4.3) leads to:

˙ε ∝ σ

(4.4)

50 Flow and fracture of a crystalline material

Time,

Strain, e

Figure 4.7. Variation of total

strain with time during

deformation of a single crystal

oriented so that glide occurs

on the basal plane (easy

glide).

at constant temperature. In other words, the velocity of dislocations varies

with σ , and their density varies with σ

, leading to a cubic dependence

of ˙ε on σ .

A large volume of experimental data on ice deformed in the labo-

ratory and on natural ice in glaciers and ice shelves yields such a cubic

dependence, particularly for stresses above 0.1 MPa. This suggests that

the theoretical model presented above is fundamentally sound, and thus

that drag mechanisms which determine the velocities of dislocations may

be the rate limiting factors.

When a single crystal of ice is stressed in such a way that there is an

appreciable component of the stress on the basal (0001) plane, the defor-

mation rate increases with time (Figure 4.7). This can be explained by the

fact that the dislocation density is normally low in unstressed crystals,

and increases gradually to its steady-state value, given by Equation (4.3),

after the stress is applied. This further supports the suggestion that drag

is the rate limiting process.

Climb as the rate-limiting process

There is, however, an additional dislocation process that must be consid-

ered. As dislocations multiply, they commonly begin to interfere with one

another, resulting in gridlock. These dislocation tangles inhibit defor-

mation. Under such conditions, application of a stress to a previously

unstrained sample results in a deformation rate that initially decreases

with time, a process called work hardening (Figure 4.8a). At sufﬁciently

low temperatures this decreasing creep rate may continue indeﬁnitely,

but at higher temperatures recovery processes come into play. One such

process involves diffusion of atoms away from (or vacancies toward)

dislocations resulting in movement of the dislocation from one crystal-

lographic plane to another. For example, if the atom at D in Figure 4.3b

diffused away, the dislocation would move upward. This is called