Hooke R.L. Principles of glacier mechanics

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Rate-limiting processes 51

Time,

Strain, e

Transient or

primary

creep

Secondary or

steady-state

creep

Ter tiary

creep

Work

hardening

(a)

(b)

Strain rate, e, a

−1

−4

−3

−2

−1

Strain,

Stress 1.02 MPa

Temperature −10

Figure 4.8. (a) A typical strain–time curve for a sample of polycrystalline ice

loaded in uniaxial compression. In early experiments, plots like this were used to

identify the time span over which steady-state creep appeared to prevail. (b) As

laboratory precision improved, it became possible to plot cumulative strain

against strain rate, and thus more accurately identify the minimum strain rate.

Note the increase in strain rate after about 1% strain. (Plot (b) is adapted with

Chemical Society.)

52 Flow and fracture of a crystalline material

dislocation climb if the dislocations are of the edge type, and cross slip

if they are of the screw type. Both processes relieve tangles and, after an

initial transient period, allow deformation to continue at a more-or-less

steady rate (central part of curve in Figure 4.8a). As climb is the recovery

process requiring more energy, it would be rate controlling.

Another recovery process, believed to be active particularly at low

stresses, involves movement of grain boundaries as some crystals grow

at the expense of their neighbors. The ice behind the moving boundary is

unstressed, and thus relatively free of dislocations; ρ

is thus decreased

by grain-boundary migration.

It is noteworthy that samples of polycrystalline ice deformed in com-

pression invariably go through a transient phase of decelerating creep and

then a period of nearly constant creep rate. (If the test is continued long

enough, the creep rate normally increases again in what is called ter-

tiary creep (Figure 4.8a); this is attributed to recrystallization, which

is discussed further below.) This supports the suggestion that recovery

processes are rate limiting. Furthermore, it may be signiﬁcant that the

activation energy for creep (79 kJ mol

−1

)isclose to that for self-diffusion

(60 kJ mol

−1

), again suggesting that climb (which is a result of diffusion)

is the rate-limiting process. However, the difference between these two

activation energies seems to be larger than can be explained by experi-

mental error so other processes may also be involved.

Slip on other crystallographic planes

Deformation of one crystal in a polycrystalline aggregate necessi-

tates deformation of neighboring crystals to preserve continuity of the

medium. Thus, all crystals must take part in the deformation. Most crys-

tals will be oriented so that the applied stress on that crystal will have a

component parallel to the basal plane. However, a few crystals will not

be so oriented, and these must therefore deform in some direction that

is not parallel to the basal plane.

Slip on either the prismatic or pyramidal planes (Figure 4.9)isa

possibility. However, on the prismatic planes the slip would probably be

parallel to the basal plane (Figure 4.9), and so would not accommodate

stresses normal to that plane. Thus, slip on pyramidal planes may be

necessary, and because such slip is likely to be much harder than that on

other planes, it could be rate limiting.

Inhomogeneous strain

The above considerations are based on the assumption of homogeneous

strain. However, in polycrystalline ice, the stress on any single crystal is

Internal stresses 53

−a

Pyramidal planes (0111) etc.

Prismatic planes (0110) etc.

Basal plane (0001)

Figure 4.9. Crystallographic

planes in a hexagonal crystal.

not equal to the stress on the bulk sample, as crystals that slip easily in one

direction but not in others will transmit the stress nonuniformly (Duval

et al., 1983). If it is assumed, instead, that the strain is inhomogeneous,

plastic strain is possible without slip on either prismatic or pyramidal

planes. This is because processes such as climb of dislocations normal

to the basal plane, grain-boundary migration, and grain translation and

rotation can accommodate stresses normal to this plane and facilitate

adjustments between crystals (Alley et al., 1997).

Internal stresses

The inhomogeneity of internal stresses in a polycrystalline sample merits

further discussion. When an ice sample is loaded, it ﬁrst deforms elas-

tically. Then, grains that are favorably oriented begin to deform by slip

on the basal plane. The load is thus transferred to grains less favorably

oriented, and high stresses can develop in such grains. As deformation

continues the inhomogeneity of the stress distribution becomes more

pronounced, with peak stresses in individual grains that may be many

times the mean stress in the sample (Duval et al., 1983).

It has been found experimentally that the resistance to shear on basal

planes in monocrystals (single crystals) is much less than the resistance

to shear on other crystallographic planes. This is because of the paucity

of bonds between basal planes mentioned earlier (Figure 4.2b). For

example, to produce a strain rate of 1.0 a

−1

at −10

◦

C requires a stress

54 Flow and fracture of a crystalline material

Monocrystals

− basal glide

Isotropic polycrystalline ice

Monocrystals

− upper bound for

non-basal glide

−10

Axial stress

MPa

−1

−2

Axial strain rate, a

−1

Figure 4.10. Stress–strain

rate data for ice at –10

◦

(Adapted with permission

from Duval et al., 1983,

American Chemical Society.)

of about 0.03 MPa in a monocrystal oriented for easy glide, and a stress

of about 3 MPa in a monocrystal oriented for hard glide (Figure 4.10).

Thus, within a sample of polycrystalline ice the local stress may vary

over two orders of magnitude. Not surprisingly, the bulk strain rate of

the polycrystalline sample lies between the values for the monocrystal

in the easy-glide and hard-glide orientations.

A consequence of this inhomogeneous internal stress distribution is

that grains that are not favorably oriented for basal glide may accommo-

date some of the plastic deformation in adjacent crystals by deforming

elastically. These crystals, therefore, have stored elastic strain energy.

Upon release of the load, this elastic energy exerts a “back stress” on

the crystals that previously deformed plastically, and a small amount

of reverse creep results (Duval, 1978)(Figure 4.11). This reverse creep

is not instantaneous, as would be the case in a purely elastic medium,

because the grains favorably oriented for basal glide under the original

loading must now creep “backwards” to relieve the stored elastic energy.

Recrystallization

Crystals of glacier ice vary in size and also in the degree to which they

are interlocked. If there were no bonding across grain boundaries, for

example, some polycrystalline ice samples would fall apart into a pile of

roughly equant grains, up to a few millimeters in maximum dimension,

while others would hang together like a three-dimensional jigsaw puzzle.

We will use the term texture to refer to these characteristics of crystal size

Recrystallization 55

−5

0102030

Strain

Time since stress reduction, min

Figure 4.11. Reverse creep due to release of stored elastic strain energy upon

unloading of a sample. The original loading of the sample was 1 MPa. At time 0,

the stress was reduced by 0.06 MPa (upper curve) and by 0.13 MPa (lower

curve), respectively. Horizontal parts of the curves reﬂect a balance between

further creep under the reduced stress and delayed release of elastic strain

energy. (Adapted with permission from Duval et al., 1983, Figure 5. Copyright

1983 American Chemical Society.)

and shape. In addition, under prolonged strain the c-axes of the crystals

develop a variety of preferred orientations, or fabrics. Both texture and

fabric affect the rheology of ice.

In order to study these processes, glaciologists, like petrologists,

use thin sections. The thin sections are typically somewhat less than a

millimeter thick and 60–80 mm across. When polarized light is passed

through a thin section and then observed through another polarizing

ﬁlter oriented at right angles to the ﬁrst, the individual crystals can be

seen because the crystal structure rotates the light as it passes through

the crystal, and the amount of rotation depends on the orientation of the

crystal. When thus viewed, the different crystals have different colors

(or grayscale tones in a black and white image – Figure 4.12). With the

use of a universal stage on which the thin section can be both rotated

around a vertical axis and tilted about either of two mutually perpendic-

ular horizontal axes, crystals can be oriented so their c-axes are vertical.

In this orientation, the crystal remains black as the stage is rotated around

its vertical axis. The orientation of the crystal is then noted and plotted

on an equal-area net (Figure 4.13). To interpret such a plot, visualize a

hemisphere with its convex side down and with a crystal in its center.

The c-axis of the crystal intersects the hemisphere. A point on a fabric

diagram like those in Figure 4.13 is the projection of this point of inter-

section onto the ﬂat surface. Thus, a vertical c-axis plots at the center

56 Flow and fracture of a crystalline material

(a) (b)

0 20 mm

Figure 4.12. Photographs of thin sections of ice from the Greenland ice sheet

near Thule. Photographs were taken under crossed polarizers. The different

grayscale tones of the grains reﬂect different orientations of the c-axes. (a) Initial

texture formed by compaction of snow with addition of small amounts of

melt water. The c-axes have a weak preferred orientation, with a preference for

vertical orientations. (b) Texture resulting from grain growth with little or no

deformation. The c-axes still have a weak vertical preferred orientation.

grayscale tone (arrows) have c-axes that are nearly parallel to one another. The

grain in the lower center is bent; in the one to left of center, distinct boundaries

have formed between parts with slightly different orientations. (d) Texture

following signiﬁcant deformation. Grains are interlocked, and c-axes have a

strong preferred orientation. (From Hooke, 1970.)

of the circle, and a c-axis dipping “south” plots between the center and

the bottom of the circle. The points are normally plotted on a Schmidt

equal-area net; this net is designed so that a unit area on the hemisphere

plots into a unit area on the net. Consequently, a c-axis dipping at 45

◦

actually plots about 55% of the distance from the center of the net to the

boundary.

Recrystallization 57

(f)

(a) (b)

(d) (e)

(g)

(c)

Figure 4.13. Examples of crystallographic fabrics observed in ice. Plots are

projections on lower hemisphere of an equal-area net. Triangles on edges show

direction of bubble elongation. In fabrics produced by simple shear, direction of

shear, shown by arrows, is presumed to be parallel to bubble elongation. All

fabrics except (c) were measured on cores from boreholes in Barnes Ice Cap

(Figure 4.15); (c) is schematic. (a) Fabric with weak preferred orientation of

c-axes in superimposed ice. (b) Fabric resulting from uniaxial compression

normal to the plane of the diagram. (c) Schematic fabric formed in pure shear.

(d) Broad single-maximum fabric. (e, f, g) Fabrics resulting from simple shear in

plane of diagram. ((a) and (d)–(g) from Hooke and Hudleston, 1980; (b) from

Hooke and Hudleston, 1981.)

58 Flow and fracture of a crystalline material

Ice that forms from compaction of snow, perhaps with some addition

of percolating meltwater, usually consists of crystals that are 2–4 mm

in diameter (Figure 4.12a). Through a series of processes that we will

refer to, collectively, as dynamic recrystallization, the texture and fabric

of this ice are altered during deformation. Dynamic recrystallization,

or simply recrystallization, is a consequence of the high local internal

stresses mentioned above, and the resulting widely differing internal

energies in adjacent grains.

One or more of three processes may be involved in recrystallization.

In order of increasing energy difference between adjacent grains, these

are grain growth, polygonization, and nucleation of new grains (Duval

and Castelnau, 1995). (Note that the terminology for these processes

differs among authors.) Grain growth results from relatively slow migra-

tion of grain boundaries. This migration is driven by the decrease in free

energy that accompanies the reduction in total area of grain boundaries

(Montagnat and Duval, 2000). Typical rates range from ∼10

−3

mm a

−1

at −30

◦

C and 1 kPa (local driving stress) to ∼10 mm a

−1

at −10

◦

C and

300 kPa (Duval et al., 1983). The migration is driven by the curvature

of the grain boundaries. Higher pressures occur on the concave sides of

such boundaries, which is commonly the side of the smaller grain, and

molecules tend to move from the high-pressure side to the low-pressure

side of the boundary (Alley, 1992). Thus, smaller crystals disappear. The

result is a characteristic texture with equant crystals of relatively uniform

size (Figure 4.12b). Because temperatures in the accumulation zones of

polar ice sheets are relatively constant to depths of a few hundred meters

(see Figure 6.6a), grain boundary migration occurs at relatively constant

rates and grain size thus increases nearly linearly with depth. At greater

depths, grain size becomes approximately constant because polygoniza-

tion, which decreases grain size, balances grain growth (Alley et al.,

1995).

Polygonization (also called rotation recrystallization)involves the

alignment of dislocations to form a new grain boundary within a bent

crystal. The crystal is thus divided into two grains with nearly the same

orientation (Figure 4.12c,arrows). Under relatively high strain rates,

polygonization begins at strains of ∼1% (Duval and Castelnau, 1995),

but at the much lower strain rates found in the central regions of conti-

nental ice sheets, cumulative strains can approach 100% without causing

polygonization (Alley, 1992). Thus, polygonization occurs at relatively

shallow depths in temperate glaciers, but is normally found only at depths

greater than a few hundred meters in polar ice sheets.

Nucleation of new grains entails the appearance of small grains

that are oriented for easy glide, with their basal planes parallel to the

maximum resolved shear stress. When they ﬁrst appear, such grains

Recrystallization 59

are relatively unstrained in comparison with adjacent older deformed

grains. As the free energy of the system can be lowered by migration

of boundaries of the new grains into the adjacent ones, the new grains

grow at the expense of the older ones (Alley, 1992). This is probably

partly responsible for the interlocking textures seen in highly deformed

ice (Figure 4.12d).

Such nucleation and grain growth (also called migration recrys-

tallization) occurs at relatively high strain rates and predominantly at

temperatures above ∼−10

◦

C. It is thus characteristic of basal ice in

temperate glaciers and of ice in the lowermost few hundred meters of

polar ice sheets.

Development of fabrics with preferred orientations

of c-axes

As just noted, newly nucleated unstrained crystals tend to grow at the

expense of older deformed ones. However, as they grow, the continued

straining gradually rotates them into orientations that are no longer opti-

mal, and they begin to accumulate strain energy. Consequently they, in

turn, are eventually consumed by even newer grains. It is the resulting

preference for orientations with basal planes parallel to the maximum

resolved shear stress that results in the increase in creep rate, in laboratory

experiments, after about 1% strain (Figure 4.8b). Preferred orientations

of c-axes are the manifestation of this preference (Figure 4.13). Thus two

basic processes are involved in the development of these c-axis fabrics:

recrystallization and grain rotation.

Let us illustrate these processes by tracing the development of such

fabrics, starting with ice near the surface in the accumulation area (Figure

4.12a). The c-axes of the crystals are either uniformly distributed or have

aweak preference for vertical orientations (Figure 4.13a). The latter

probably results from the orientation of snowﬂakes that have thin disk-

like shapes, and thus, like a pile of poker chips, tend to lie ﬂat as they

accumulate. In addition, the vertical temperature gradient may have had

some inﬂuence during sintering.

As the ice becomes buried, it is compressed vertically and stretched

longitudinally and sometimes also laterally. Where longitudinal and lat-

eral strain rates are comparable in magnitude, the stress ﬁeld is referred

to as uniaxial compression (Figure 4.14a), whereas if lateral strain rates

are negligible it is pure shear (Figure 4.14b). In such stress conﬁgu-

rations, slip occurs most readily on the basal planes of crystals whose

c-axes are inclined at ∼45

◦

to the compression axis (Figure 4.14d).

Thus crystals are nucleated in this orientation, and these crystals grow at

the expense of adjacent more highly stressed ones, leading to a conical

60 Flow and fracture of a crystalline material

Uniaxial

compression

(a)

Resolved

shear

(b)

Pure shear

Compression

Extension

(c)

Simple shear

(d)

Pure shear

Compression

Extension

Becomes

∼

–

after rotation

(e)

(f)

Figure 4.14. Stress conﬁgurations and their relation to orientations of c-axes.

(a) Uniaxial compression. (b) Pure shear. There is no strain normal to the plane

of the diagram. (c) Simple shear. (d) The c-axes and basal planes in a newly

nucleated crystal in uniaxial compression or pure shear. With continued

compression, the c-axes rotate toward the axis of compression. (e) Simple shear

viewed parallel to the shear direction with basal planes also parallel to the shear

direction as in the fabric of Figure 4.13f. The symbols

and signify stress

vectors directed into and out of page, respectively. (f) Simple shear viewed

normal to the shear direction with basal planes inclined to the shear direction as

in the fabric of Figure 4.13g.In(d)–(f) short-dashed lines show orientations of

basal planes.

distribution of c-axes in uniaxial compression (a small-circle fabric:

Figure 4.13b.) and to two maxima aligned in the direction of extension

in pure shear (Figure 4.13c). (Small-circle fabrics are also commonly

referred to as girdle fabrics, although “girdle” implies a great circle.)

The vertical compression and lateral extension, however, have the effect