Hooke R.L. Principles of glacier mechanics

Подождите немного. Документ загружается.

Simple and pure shear 351

A useful measure of the magnitude of the total strain experienced by the

ellipsoid is then:

[

(

− ε

)

(

− ε

)

(

− ε

)

]

1/2

(13.4)

is called the natural octahedral unit shear (Nadai, 1950,p.115) and the

additional subscript c denotes cumulative. (“Natural” in this context is a

reference to the appearance of the natural logarithm in the expressions for

, not to a contrast with some “artiﬁcial” counterpart.) The expression

on the right-hand side of Equation (13.4)istwice the root mean square of

the strains in the three mutually perpendicular planes of our coordinate

system.

In plane strain with ε

= 0 and ε

=−ε

, the intermediate or

Y-axis of the ellipsoid will be transverse to the ﬂow and thus parallel

to the y-axis of the ﬁxed coordinate system. Equation (13.4) can then be

simpliﬁed and combined with Equation (13.2)toyield:

√



(13.5)

or using Equation (13.2)toeliminate 

√



(13.6)

Simple and pure shear

As we have discussed previously (Figure 4.14), when a two-dimensional

object is deformed by a compressive stress parallel to the z-axis and

an extensional stress parallel to the x-axis, the deformation is known as

pure shear (Figure 13.3a), whereas if it is deformed by a shear stress,

, parallel to the x-axis, the deformation is known as simple shear

(Figure 13.3b). In Figure 13.3a the axis of maximum shortening is clearly

parallel to the z-axis and the axis of maximum extension is parallel to

the x-axis. This is also true of the instantaneous stretching axes,oraxes

along which the maximum and minimum normal strain rates occur dur-

ing any inﬁnitesimal strain. Material lines (lines of physical points in the

deforming material) that are initially parallel to the instantaneous stretch-

ing axes remain parallel to these axes. The strain is thus irrotational, and

the deformation is said to accumulate coaxially.

In contrast, in simple shear the instantaneous stretching axes are

always at ±45

◦

to the direction of shear (θ

= 45

◦

,where the subscript

“in” refers to instantaneous or inﬁnitesimal; see Figure 13.3b). How-

ever, the axis of maximum extension will be at 45

only after the ﬁrst

inﬁnitesimal increment of strain. In the next increment, the initial axis

352 Finite strain and the origin of foliation

(b)(a)

Simple shear

Pure shear

Figure 13.3. Sketches illustrating (a) pure and (b) simple shear.

of maximum extension will be rotated slightly. This axis is parallel to a

material line that started at an angle greater than 45

◦

to the x-axis and

that has been rotated through the 45

◦

position to an angle less than 45

◦

This is most easily visualized if one considers very large strains, as in the

lower right part of Figure 13.1. The material line that became the nearly

horizontal axis of maximum extension in this case started off nearly

vertical. Because material lines rotate through the axes of instantaneous

stretching (which are non-material lines) the deformation in simple shear

is rotational and non-coaxial.

In either simple or pure shear there are always two lines that are

initially perpendicular and that are still perpendicular after deforma-

tion. These lines coincide with the directions of maximum extension (or

elongation) and shortening in the deformed state. They are called the

principal axes of strain.

Parameters describing cumulative deformation

Although γ

describes the deformation of the original sphere, it does

not describe either its rotation or its ﬁnal orientation. For that we need

two additional parameters, θ and ϕ (Figure 13.2); ϕ

is the angle through

which the material line that becomes a principal axis in the strained state

has rotated, and θ

is the angle that the greatest principal axis makes

with the x-axis.Again, the subscript c denotes cumulative.

In the pure shear of Figure 13.3a, the axis of maximum elongation,

the X-axis of the ellipsoid, is horizontal, both for inﬁnitesimal strains

and for cumulative strain over a long period of time. Thus, θ

is 0, and

because the X-axis does not rotate, ϕ

is also 0. In contrast, in simple

shear as in Figure 13.3b, the X-axis is at 45

◦

to the shear direction for any

inﬁnitesimal increment of strain. However, during any such increment,

the line that became the X-axis will have rotated slightly from its original

orientation, so ϕ

is slightly greater than 0 and θ

slightly less than 45

◦

Calculating cumulative strain 353

Equilibrium

line

Mixed

Simple shear

Pure shear

Particle path

Isotime line (parallel to stratification)

Divide

Figure 13.4. Schematic illustration of the strain ﬁelds in an ice sheet. The ﬁelds

are for small increments of strain only. The horizontal component of the pure

shear ﬁeld is extensional above the equilibrium line and compressive below.

(From Hooke and Hudleston, 1978, Figure 3A. Reproduced with permission of

the International Glaciological Society.)

As the strain accumulates, ϕ

will increase toward 90

◦

(but it will never

actually become 90

◦

) and θ

will approach 0

◦

. The reader may wonder

why ϕ

→ 90

◦

when the X-axis after the ﬁrst increment of strain was

inclined at 45

◦

. The answer lies in the phrase “...the rotation of the

line that becomes the principal axis . . .”. After very large shear strains,

the line that becomes the X-axis will be the one that started out nearly

vertical.

As one would expect from consideration of velocity proﬁles like

those in Figure 10.6, pure shear dominates near the glacier surface where

∂u/∂x and ∂w/∂z are large compared with ∂u/∂z, and conversely, simple

shear dominates near the bed where ∂u/∂z becomes large compared

with ∂u/∂x and ∂w/∂z (Figure 13.4). Immediately beneath the divide,

∂u/∂z = 0atall depths, so the pure shear ﬁeld extends all the way

to the bed. Near the equilibrium line, the vertical velocity decreases

and the ﬂow gradually changes from extending to compressive so both

∂w/∂z and ∂u/∂x decrease. Thus the ratio of simple shear to pure shear

increases, so despite the small ∂u/∂z in the upper part of the glacier the

thickness of the pure shear ﬁeld decreases.

Calculating cumulative strain

In order to calculate the cumulative strain in a glacier, one ﬁrst must

know the velocity ﬁeld. One then calculates the path that a particle of

ice would follow through the glacier and velocity derivatives at discrete

points along the path. To obtain the incremental strain as the ice moves

from one point to the next, strain rates are then calculated from the

velocity derivatives and multiplied by the time needed for this movement

354 Finite strain and the origin of foliation

500

1000

2000

4000

1000

2000

4000

Particle path Isochron

T020

T061

T081

Divide

and isochrons

(a)

Particle paths

(b)

Direction of maximum cumulative extension

(c)

Cumulative strain magnitude,

0.5

01 2km

200

Figure 13.5. (a) Particle paths calculated from the velocity ﬁeld described in

the text. Ages of isochrons are shown in years. (b) Orientation of maximum

cumulative extension direction shown by bars at 200 m intervals along the

particle paths in (a). (c) Contours of

. (From Hudleston and Hooke, 1980,

Figures 6 and 7. Reproduced with permission of Elsevier Scientiﬁc Publishing.)

(Ramsay and Graham, 1970, Equations 7–10). Finally, these incremental

strains are added to get the cumulative strain.

Hudleston (Hooke and Hudleston, 1980; Hudleston and Hooke,

1980) made such a calculation using a comprehensive set of data on

velocities and mass balance along the ﬂow line on Barnes Ice Cap illus-

trated in Figure 13.5a (see also Figure 12.17). To estimate horizontal

velocities at depth, he used measured surface velocities and adjusted

the value of B in Equation (5.7)toget zero velocity on the bed, where

the temperature is well below the melting point. Rather than use mea-

sured vertical velocities, he used the mass balance data to estimate the

long-term steady-state vertical velocity at the surface and assumed that

it decreased linearly with depth. Transverse strain rates are small, so he

assumed that they could be neglected.

The results of his calculations are shown in Figures 13.5 and 13.6.

By following particles starting at nine points in the accumulation area,

he ﬁrst mapped nine ﬂowlines (Figure 13.5a). He then calculated the

Calculating cumulative strain 355

(a)

(b)

Depth, m

Figure 13.6. Calculated variation of γ

,yiθ

, and ϕ

with depth at locations of

Barnes Ice Cap boreholes (a) T020 and (b) T081. Locations of boreholes are

shown in Figure 13.5. (From Hooke and Hudleston, 1980, Figures 8a and 9a.

Reproduced with permission of the International Glaciological Society.)

orientations of the axes of maximum cumulative extension (X-axes:

Figure 13.5b). Noteworthy in this ﬁgure is the fact that these axes are

nearly parallel to the bed throughout most of the glacier. The increased

upglacier dip at the surface near and downglacier from borehole T061 is

a consequence of the increase in the ratio of simple shear to pure shear

as the equilibrium line is approached. As just noted, in simple shear the

axis of maximum elongation dips 45

◦

initially; with increasing defor-

mation it is gradually rotated toward parallelism with the plane of the

shear.

The cumulative strain magnitude,

,isshown in Figure 13.5c.

These numbers do not appear signiﬁcant until one realizes that

proportional to the natural logarithm of the axial ratio of the strain ellipse.

Thus,

= 8, found in the most basal ice, corresponds to an elongation

of ∼18 000:1.A1mcube would be stretched into a 1-m wide ribbon

134 m long and 7.5 mm thick!

Figure 13.6 shows the variation of

, θ

, and ϕ

with depth in

boreholes T020 and T081 (Figure 13.5a). Because the dominant strain

pattern at T020, particularly in the upper part of the glacier, is nearly pure

shear with vertical compression and longitudinal extension, the axis of

maximum cumulative extension is nearly horizontal. Thus, θ

remains

close to 0. On the other hand, ϕ

is 0 at the surface and initially increases

356 Finite strain and the origin of foliation

gradually with depth as ∂u/∂z increases. However, with increasing depth

in the glacier, the ice arriving at T020 has passed through a larger and

larger region dominated by pure shear (Figure 13.4). Thus, ϕ

reaches

a maximum (∼12

◦

)atadepth of ∼240 m and then decreases at greater

depth.

The pattern at the site of borehole T081 is different in several

respects. Because this hole is in the upper part of the ablation area, ice

at the surface accumulated some strain as it moved from higher in the

glacier. Thus,

> 0atthe surface. With increasing depth, the ice has

traveled a greater distance and accumulated more strain so

increases

to about 7, representing an axial ratio of over 5000. Because ice near

the surface has experienced a modest amount of mixed simple and pure

shear (Figure 13.4), θ

≈ 10

◦

and ϕ

≈ 25

◦

here. With increasing depth,

ﬁrst increases, reaching a maximum at a depth of ∼120 m and then

decreases, reﬂecting the early history of pure shear that this deeper ice

experienced.

To further quantify the inﬂuence of the early history of pure shear,

Hudleston calculated ϕ

for a particle of ice that experienced a total strain,

,of3.75 entirely by simple shear (Hooke and Hudleston, 1980). In

this case, ϕ

is 80

◦

and increases toward 90

◦

as γ

increases further. For

comparison, in holes T020 and T081 the actual rotations at this strain

magnitude are 9

◦

and 57

◦

, respectively.

Let us now use our understanding of cumulative strain to study the

origin of foliation.

Components of foliation

The pronounced banded character of glaciers (see, for example, Figures

5.18 and 8.8) has led to considerable confusion. Banding is most promi-

nent in the ablation area once the winter snow has melted. However, band-

ing may also be seen in crevasse walls in the accumulation area, although

it has a very different appearance there and most people would, correctly,

refer to it as annual layering or sedimentary stratiﬁcation. The banding

is normally subparallel to the nearest bounding surface, be it the bed, the

surface, or the valley walls. However, in the lower part of the ablation area

it typically dips gently to steeply upglacier (Figure 13.7b). The banding

is penetrative; that is, the bands are cross sections of layers in the ice.

On close inspection, one ﬁnds that the banded appearance most com-

monly results from variations in bubble or dirt content. The latter, cou-

pled with the suggestive upglacier dip of the bands near the margin, has

given rise to the mistaken impression that the banding is a reﬂection of

shear planes in the ice along which debris was (somehow) carried to the

Components of foliation 357

(a)

100200300

Boreholes

Distance from margin, m

Ice tunnel

Trace and measured dip of foliation

(b)

Figure 13.7. (a) Foliation attitudes in the margin of an idealized perfectly

plastic glacier with a parabolic surface proﬁle. See text for explanation. (b) Cross

section of the margin of Barnes Ice Cap showing dips of foliation planes

measured on the ice surface and in a tunnel excavated in the margin. Dashed

lines show traces of foliation planes inferred from measured dips and core

stratigraphy.

surface. This is rarely the case. Elongate bubbles and, in some instances,

elongate crystals may contribute to the banded appearance.

Foliation is the end product of subjecting inhomogeneities of any

origin to cumulative stretching and shearing by strains such as those

discussed in the ﬁrst part of this chapter. After such treatment, the origin

of the initial inhomogeneity is not likely to be obvious. Suppose, for

example, that the 1 m cube we mentioned earlier were a Michelangelo

sculpture in a material with a rheology similar to that of ice. Would we

recognize it after it had become a ribbon 135 m long and 7.5 mm thick?

Perhaps we could identify the material, but certainly not the original

handsome shape.

358 Finite strain and the origin of foliation

So what are the inhomogeneities that result in foliation? The most

obvious is the sedimentary stratiﬁcation that one sees in crevasse walls

in the accumulation area. This stratiﬁcation is deﬁned by variations in

grain size, particularly near the base of an annual layer where depth

hoar forms. Wind-blown dust picked up from snow-free ground dur-

ing the melt season commonly discolors the surface of annual layers,

and percolating meltwater may spread out and refreeze along horizons

deﬁned by variations in snow density, forming relatively bubble-free

lenses. Both contribute to the stratiﬁed appearance. Gland-shaped bod-

ies of ice may also form from refreezing of percolating meltwater (p. 20).

From the pattern of elongation directions shown in Figure 13.5b,itis

clear that these inhomogeneities will be stretched longitudinally and

compressed vertically. Thus, the deformation accentuates the original

layering. By the time the inhomogeneities reappear low in the abla-

tion area, they will bear little resemblance to the original sedimentary

stratiﬁcation or glandular structures. Once such inhomogeneities have

been deformed beyond ready recognition, the structure is properly called

foliation.

Another major component of foliation is crevasse ﬁllings. Snow

blows into crevasses during the winter and meltwater may saturate this

snow during the summer. When the crevasse gets moved into a location,

or rotated into an orientation, such that stresses across it are compres-

sional, it closes and this ﬁlling gets squashed. A band is thus formed that

cross cuts either sedimentary layering or foliation derived from such lay-

ering. Initially, this cross-cutting relation is obvious and the ﬁlling can be

properly identiﬁed as what it is, even if it cross cuts foliation the origin of

which cannot be identiﬁed. However, once the ﬁlling has been squashed

further, rotated to near parallelism with other foliation, and stretched, its

origin will be less obvious. Careful observers, however, may still be able

to detect the cross-cutting relation.

Another prominent component of foliation in basal ice is debris. As

we have discussed, there are various mechanisms by which debris may

become entrained in basal ice – the refreezing part of the regelation

cycle, freeze-on by the mechanisms discussed by Robin (Figure 7.6)or

in areas where temperature gradients in basal ice lead to local refreezing

(Figure 6.12), entrainment in frazil ice in overdeepenings (Figure 8.33),

and so forth. In all situations, the resulting layers are parallel to the bed

or valley walls and are in areas subject to high rates of simple shear. The

precise origin of the layers is hard to determine before they become

deformed, and after shearing the task is usually hopeless, although

chemical and isotopic signatures may help (Souchez and Lorrain,

1978).

Components of foliation 359

Foliation in the marginal zone

We noted above that foliation in glacier margins has often been mistaken

for so called “shear planes” that would presumably be parallel to planes

of maximum shear stress. Here, we demonstrate that the upglacier dip

of the foliation at a glacier surface bears little relation to directions of

maximum shear stress.

As in the case of Figures 13.5 and 13.6, the ﬁrst task is to determine

the velocity ﬁeld. Then we use that velocity ﬁeld to see how foliation

that is nearly parallel to the bed some distance from the margin will be

deformed as it is passively advected to the margin.

We adopt a coordinate system in which the origin is at the margin,

the x-axis is horizontal and directed upglacier along the bed, and the

z-axis is vertical and positive upward (Figure 13.7a). We assume that:

(1) the proﬁle of the ice surface in the marginal zone can be approximated

by h =

√

cx,where h is the surface elevation, x is the distance from the

margin and c is a constant

; (2) the ablation rate, b

(a negative number),

is uniform over the area of interest; (3) the proﬁle is a steady-state proﬁle

with b

=−w

+ u

tan α (Equation (5.26)); (4) the horizontal velocity,

u,isindependent of depth; (5) the strain is two dimensional (plane

strain); and (6) ice is incompressible. The amount of ice lost by ablation

downglacier from a position x is then b

x and this must equal the mass

ﬂux past x,or

uh. Thus,

u =

√

= x

1/2

(13.7)

where  =b

1/2

. (As b

is negative,  is negative. This is consistent with

u being in the negative x-direction.) The assumption of incompressibility

may be represented by:

∂u

∂x

∂w

∂z

= 0 (13.8)

Differentiating Equation (13.7) and using the result to integrate Equation

(13.8) yields:

w =−

x

−1/2

z (13.9)

We now determine the coordinates of an element of ice as a func-

tion of time. The element starts at position (x

, z

)attime t = 0. As

u = dx/dt, Equation (13.7) can be integrated to obtain:

x =



1/2

t



(13.10)

This equation, ﬁrst derived by Nye (1951), can be obtained by integrating Equation (5.2c)

with α = dh/dx. The reader will then ﬁnd that c =

√

2σ

/ρg.

360 Finite strain and the origin of foliation

Then combining Equations (13.9) and (13.10)toeliminate x, noting that

w = dz/dt, and integrating we obtain:

z = z





1/2

(13.11)

where z is the height of the element of ice after it has traveled a distance

– x) from (x

, z

). The position of the element at time t

can be

determined by selecting a time t

> 0 and solving Equation (13.10) for

x and then Equation (13.11) for z.Aplot of x versus z deﬁnes the path

of the element. By setting z =

√

cx in Equation (13.11) and solving

for x we can determine the x-coordinate of the point where the element

reaches the glacier surface.

Now consider a foliation plane in this element of ice. At (x

, z

) the

plane has an upglacier dip, β

, with respect to the particle path, and we

wish to compute its dip when the element reaches the glacier surface.

(Because θ

is always greater than 0 under steady-state conditions, the

foliation plane will dip more steeply than the particle path. The conse-

quences of a non-steady-state situation will be discussed further below.)

The element is subjected to longitudinal compression and vertical exten-

sion which will change the inclination of any line that is not parallel to

the direction of compression or extension (Figure 12.12). In addition,

at any given height above the bed, w increases toward the margin as h

decreases (Equation (13.9), noting that x

1/2

= b

/h). Therefore, the

shear strain ∂w/∂x will also increase the upglacier inclination of the

plane. The problem thus strongly resembles the deformation of a bore-

hole (Chapter 12)except that now the “hole” is nearly horizontal, not

vertical, and the primary shear to which it is subjected is ∂w/∂x, not

∂u/∂z. Consequently, Equation (12.49)isapplicable in the form:

∂β

∂t

∂w

∂x

+ 2β

∂w

∂z

− u

∂β

∂x

(13.12)

To express β in terms of x alone, we obtain ∂w/∂x and ∂w/∂z from

Equation (13.9) and use the chain rule:

dβ

to evaluate the term on the left. Noting that dx/dt=u and using Equation

(13.11), we end up with a ﬁrst-order linear ordinary differential equation:

dβ

−

1/2

5/2

= 0 (13.13)

which can be solved by setting β =a(x)b(x), where a and b are functions

of x to be determined (Sokolnikoff and Redheffer, 1958, pp. 23–24). The

result is:

β =







−



(13.14)