The Pennsylvania State University
College of Earth and Mineral Sciences
Department of Geosciences
A MODEL OF HORIZONTAL FRACTURE
PROPAGATION AS A RATE-LIMITING PROCESS IN
GLACIER EROSION
A Thesis in Geosciences
by
Timothy T. Creyts
Submitted in Partial Fulllment
of the Requirements
for the Degree of
Bachelor of Science
May, 1998
We approve this thesis:
Richard B. Alley
Professor of Geosciences
Thesis Advisor
Date
Earl K. Graham
Professor of Geophysics
Associate Head for the Undergraduate Program
Date
ii
I grant The Pennsylvania State University the nonexclusive right to use this work for the
University's own purposes and to make single copies of the work available to the public
on a not-for-prot basis if copies are not otherwise available.
Timothy T. Creyts
Abstract
iii
In glaciated, mountainous regions, glaciers erode large amounts of rock. Previous researchers have used fracture mechanics to demonstrate that erosion rates vary by
orders of magnitude for subglacial water-pressure variations within the natural range.
They assumed that vertical, subcritical crack-growth is the rate-limiting step in glacier
erosion. In sedimentary rock or in crystalline rock with well-developed, horizontal fractures, vertical crack-growth must be rate-limiting. However, in other rocks horizontal
fracture growth may be rate-limiting.
Horizontal fracture propagation depends on two superposed stresses: that from
changes in subglacial hydrology, and the near-surface stress. In order to assess the role of
horizontal fracture growth, two separate numerical models were used. The rst computes
the near-surface stresses while the second model computes the hydrologic stresses. The
model results indicate that the near-surface stress acts to align the crack-propagation
direction horizontally, while the daily cycles in the hydrologic stress fatigue the rock and
induce fracturing. It is also found that the number of cycles necessary to fatigue the
bedrock varies considerably (<10,000->5,000,000). This variation means that horizontal
fractures require <100 to >50,000 years to propagate. In addition, the horizontal fractures
necessary to produce smaller rock fragments require less time and less stress variation.
Therefore, the production of smaller-sized clasts (0.1-0.4 m) is favored by the fracture
process. Because cobbles and small boulders are the most common clast sizes in glacial
sediments, one must conclude that horizontal fracture growth exists as a rate-limiting
step in glacier erosion only when cobble-scale horizontal fractures do not exist in the
bedrock beneath a glacier.
Table of Contents
iv
Abstract : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
iii
List of Tables : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
vi
List of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
vii
Acknowledgments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii
Chapter 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
1
1.1 The importance of glacier erosion . . . . . . . . . . . . . . . . . . . .
1
1.2 Objective of this study . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Chapter 2. Previous Work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5
2.1 Vertical Fracture Propagation beneath Glaciers . . . . . . . . . . . .
5
Chapter 3. Subglacial Stress Theory : : : : : : : : : : : : : : : : : : : : : : : : :
11
3.1 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2 Subglacial Channel Stress . . . . . . . . . . . . . . . . . . . . . . . .
12
3.3 Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Chapter 4. Methods and Results : : : : : : : : : : : : : : : : : : : : : : : : : : :
16
4.1 Crack Propagation Vectors . . . . . . . . . . . . . . . . . . . . . . . .
16
4.2 Fatigue Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.2.1 Model Code and Parameterizations . . . . . . . . . . . . . . .
21
v
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Chapter 5. Discussion and Conclusions : : : : : : : : : : : : : : : : : : : : : : :
26
References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
29
vi
List of Tables
Table
Page
4.1 Model Parameters for Each of the Four Variables . . . . . . . . . . . . .
22
List of Figures
Figure
vii
Page
1.1 Erosion Rates for Glaciated Basins . . . . . . . . . . . . . . . . . . . . .
3
2.1 Subglacial Geometry from Hallet . . . . . . . . . . . . . . . . . . . . . .
6
2.2 Results of Hallet's Model . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.1 Geometry of a Hertzian Indentor . . . . . . . . . . . . . . . . . . . . . .
15
4.1 Crack Propagation Vectors resulting from an Overpressured Channel on
Bedrock without a Near-surface Stress . . . . . . . . . . . . . . . . . . .
17
4.2 Crack Propagation Vectors resulting from an Overpressured Channel on
Bedrock with a Near-surface Stress . . . . . . . . . . . . . . . . . . . . .
18
4.3 Crack Propagation Vectors Resulting from a Larger, Overpressured
Channel on Bedrock with a Near-surface Stress . . . . . . . . . . . . . .
20
4.4 The Relationship of the Background Stress to the Critical Stress Intensity
Factor via the Number of Cycles to Fatigue . . . . . . . . . . . . . . . .
23
4.5 Cycles to Failure as a Function of Depth and Critical Crack Length (Clast
size) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Acknowledgments
viii
The author wishes to thank Richard Alley for his tireless enthusiasm and support. Fracture mechanics advice was graciously volunteered by Dr. Terry Engelder. The
author also wishes to thank Dr. Shelton Alexander and Joe Schall for their review of this
manuscript. Advice, as well as an occasional deaf ear, was provided by the ice groupies,
particularly Byron Parizek, Todd K. Dupont, and Sarah Das. The author would like
to thank Mark Newey, Phil Dennison, Kevin Myer, Steve Strum, and, especially, Kim
Comstock for their unceasing moral support.
This work was supported by The College of Earth and Mineral Sciences, the
Geosciences Department, The Schreyer Honors College, and NSF OPP Grant number
96-14927.
1
Chapter 1
Introduction
1.1 The importance of glacier erosion
Typical geomorphic features in glaciated areas include jagged ridges, U-shaped
valleys, and extensive moraines. These features are all products of glacial erosion, either
directly or indirectly. For example, jagged ridges would not appear jagged without the
removal of the adjacent bedrock. U-shaped valleys would not exist without glacial erosion.
Most moraines could not exist without the erosion of the glacier's subjacent bedrock
producing small fragments that could be deposited later. Therefore, glacier erosion is an
integral part of landscape evolution in glaciated areas.
Ironically, glacial erosion also may increase the height of mountain ranges. Molnar
and England (1990) proposed that the removal of bedrock in valleys by glaciers induces
uplift. This uplift is a result of isostatic rebound as the load on the mantle is lessened
(Molnar and England, 1990). Hence, mountainous, extensively glaciated terrains may be
the result of a positive feedback loop. Concentration of glacier erosion in valleys causes
uplift, which increases snowfall on the peaks and so intensies glaciation and glacial
erosion.
Another impact of glacier erosion is that it may play a key role in global biogeochemical cycles. Researchers (e.g. Kump and Alley, 1994: Gibbs and Kump, 1994)
have hypothesized that the recent decrease in global carbon dioxide concentration may
2
be attributable to enhanced glacial erosion in mountainous regions. As eroded rock is
transported away from the mountainous regions, these sediments weather chemically.
The alteration processes incorporate carbon dioxide into the weathering products of the
sediments.
1.2 Objective of this study
Typically, researchers have thought of glaciated basins as having a higher erosion
rate normalized over the area of the basin than non-glaciated basins (e.g. Slaymaker,
1987). Hallet et al. (1996) showed that warm, wet, extensively glaciated basins have
erosion rates one to two orders of magnitude higher than other glaciated basins. Figure
1.1 illustrates that glaciers from Southeast Alaska, which have a characteristic high
water-pressure variability, have considerably higher erosion rates than glaciers from other
areas which, in turn, erode as rapidly or more rapidly than do streams in similar, but
nonglaciated basins. From Figure 1.1, the average of the glacial erosion rates, excluding
glaciers from Southeast Alaska, is approximately 1 mm yr;1, whereas the average
erosion rate of glaciers from Southeast Alaska is about 24 mm yr;1.
Hallet (1996) also constructed a simple, theoretical model of glacier erosion
which demonstrates that bedrock removal varies signicantly with the water-pressurevariability of glaciers. In this model, Hallet assumed that vertical, subcritical crack
growth orthogonal to the direction of ice ow is the rate-limiting process in glacier
erosion. However, in order to erode rock, two other types of fractures need to exist|a
vertical fracture-set parallel to ow, and a horizontal fracture. It is the objective of this
study to examine the process of horizontal fracture propagation beneath glaciers through
3
Fig. 1.1. Average erosion rates for glaciated basins. The data from southeast Alaska
remain separate from the global data set. (Data from Hallet et al., 1996)
4
the use of a simple theoretical model.
5
Chapter 2
Previous Work
2.1 Vertical Fracture Propagation beneath Glaciers
Both Iverson (1991) and Hallet (1996) have developed quantitative theoretical
models of glacier-erosion. Iverson's model consists of a stair-step in the glacier bed with
the glacier owing over the step to create a cavity. Dierences in the stress regime in the
rock resulting from the uid pressure in the cavity, the ice overburden pressure, and the
basal shear stress cause fracture initiation.
Hallet used a similar sawtooth bedform (Figure 2.1) to achieve the same result.
He derived the stress balance in the horizontal and vertical directions:
+ Pw S 0 tan n (1 S 0 ) tan = 0
(2.1)
Pw S 0 + (1 S 0 )n Pi = 0
(2.2)
;
;
;
;
where is basal shear stress in the direction of ice ow, n is stress normal to the rock-ice
contact, Pw is the water pressure in the cavity, Pi is the ice overburden pressure, S 0 is
the normalized extent of ice-bed separation, S=L, with S the cavity length and L the
length for the entire bedform, is the dip of the cavity roof parallel to ice ow, and is the slope of the rock ledge.
Hallet then solved equations (2.1) and (2.2) for n in terms of Pe = Pi Pw , the
;
6
Fig. 2.1. (a) Geometry of idealized bed for the quarrying model. (b) Stresses and location
of probable fracture initiation near the corner of a ledge. Quarrying takes place when
fractures propagate downward from the ice-loaded surface and coalesce with the planar
discontinuities that dene that ledge system, at which point a bedrock block is liberated
and the ledge is propagated up-glacier (Modied from Hallet, 1996)
7
eective pressure, Pi , Pw , and S 0 :
0
n = Pi + Pe S 0
1 S
!
;
(2.3)
Hallet also noted that the dierence between the normal stress and the water pressure is
the fracture-driving deviatoric stress. He used equation (2.3) to solve for the deviatoric
stress:
0
n Pw = Pe S 0
1 S
!
;
;
(2.4)
From equations (2.3) and (2.4), one can make several important conclusions about the
water pressure:
Higher water pressures lead to larger water cavities.
Larger cavities lead to a smaller ice-rock contact surfaces.
Smaller ice-rock contact surfaces concentrate the normal stress.
In order to obtain an erosion rate, Hallet combined equations (2.3) and (2.4)
with linear elastic fracture mechanics. Specically, Hallet solved for the stress-intensity
factor and used that result to solve for the subcritical crack-growth rate. The fracture
8
mechanics equations Hallet (1991) used are:
KI =
c =
4c 12
T
2 0 0 2
K
6
B
B
I 4exp @ @ 2I
K
c = 0
Ic
KI
1K
3 Ic
;
11
1C
AC
A
;
3
8 7
exp 9 5 ;
(2.5)
KI > 13 KIc (2.6)
(2.7)
where KI is the opening-mode stress-intensity factor, c is the original penny-crack half
length, T is the far eld stress, c is the subcritical crack growth rate, I and are material parameters, and KIc is the critical KI value needed for dynamic crack propagation.
The value for KIc was from Walder and Hallet (1985), based on a review of previous
work, and it lies between 1 and 2 MPa m1=2.
Figure 2.2 depicts the results of Hallet's model. From these results, one can make
conclusions about the quarrying rate. For a given sliding velocity, at relatively high
eective pressures, the quarrying rate decreases for increasing eective pressures, because
the normal stress is distributed over a larger bed area. Conversely, for lower eective
pressures, the quarrying rate decreases with decreasing eective pressure, because of the
decrease in clast size eroded.
It is highly unlikely that subglacial conditions will fall near the peak-erosion conditions of Figure 2.2, because the peaks are quite narrow compared to the possible range of
subglacial conditions. Rather, almost all temperate glaciers will fall in one of the \ank"
regions, for which erosion rate varies exponentially with eective pressure. Eective pressure varies principally with water pressure, as the ice overburden, Pi , varies much more
9
Fig. 2.2. Calculated index of quarrying rate as a function of sliding velocity and basal
eective pressure. Model parameters were ledge width: L = 10 m treadslope: sin =
0:2 and cavity closure-rate constant: 5 a;1 MPa;3 , which reects standard rheological
parameters. Empirical fracture-mechanical data were used for: (a) Westerly Granite with
20 mm cracks, and (b) St Pons Marble with 10 mm cracks (Hallet, 1996, Fig.2.)
10
slowly than the water pressure. For any system in which the rate varies exponentially with
some controlling variable, the average rate increases with increased variability about the
mean value of that controlling variable. Hence, one can conclude from Hallet's analysis
that erosion in glaciated landscapes is directly related to the water{pressure variability
beneath a glacier. Similarly, Iverson (1991) concluded that the subglacial hydrology is
essential to glacier erosion.
11
Chapter 3
Subglacial Stress Theory
Both Hallet (1996) and Iverson (1991) assumed that surface-parallel fractures
exist such that the vertical fracturing process is the rate-limiting factor. Clearly, in wellbedded sedimentary strata, this must be the case. However, sheeting joints in igneous and
metamorphic rocks may not be inherent to the rock. Here, a model for two-dimensional,
glacier-induced horizontal fractures is presented.
3.1 Fracture Mechanics
In two dimensions, the opening-mode fracture of rock depends upon the magnitude of a tensile stress, the crack geometry, and the size of the crack. Atkinson (1984)
gave the equation
1
KI = Y r (c) 2
(3.1)
where KI is the opening-mode stress intensity factor, Y is a geometrical term, r is a
tensile stress, and c is a crack half-length. When the crack is assumed to be penny-shaped,
Y is 1. Note that this equation diers from equation (2.5) by a small numerical factor,
which Hallet used, but maintains the same functional dependence on the controlling
variables.
The stress-intensity factor (SIF) is essentially a way to quantify the strength of a
12
rock relative to its microaws and an applied stress. It is important to realize that twodimensional fracture does not simply depend on stress. For example, if a hypothetical
material existed without microaws, it would not break in typical models.
A second mechanism for fracture propagation is fatigue crack extension. In this
case, stress oscillates between two values. Paris (1964) gave a fatigue relation as
da = C (K )m
I
dN
(3.2)
where a is the crack length, N is the number of cycles, C and m are constants, and KI
is the maximum change in stress intensity over one cycle with KI given by equation (3.1).
Subsequently, Kim and Mubeen (1981) performed a series of experiments showing that
Paris' equation (3.2) ts empirical data extremely well for fracture growth in Westerly
Granite. Results of their study indicate that m varies between 11.8 and 11.9 and that C
ranges from 2 10;10 to 8 10;10 m1=2 MPa;1 . Scholz and Kranz (1974) and Scholz and
Koczynski (1979) demonstrated that recovery from fatigue decreases signicantly after
several loading cycles.
3.2 Subglacial Channel Stress
Because the maximum strength of ice corresponds to a deviatoric stress of about
6 MPa (as reviewed by Hallet, 1996), which is well below the maximum strength of rock,
erosion must take place at stresses below this deviatoric stress of 6 MPa. Therefore,
the eective pressure, Pe must also remain below 6 MPa. Hallet (1996) demonstrated
that erosion occurs below an eective pressure of 6 MPa. However, he considered only
13
subcritical crack growth and did not include the additional contribution to erosion from
fatigue associated with subglacial hydrologic changes.
Many workers (e.g. Hubbard et al. (1995) Murray and Clarke (1995)) have shown
that the water pressure of glaciers with abundant surface melting varies on a daily cycle
in the summer months. Surface melt feeds basal channels. Filling and draining of the
channels in response to the diurnal increase and decrease of melting cause pressure
uctuations. The creep of the channels closed is su ciently slower that channels remain
open at night (R!othlisberger, 1972).
Murray and Clarke (1995) used three boreholes to demonstrate that water pressure varies depending on the distance from a subglacial channel. They showed that an
overpressured channel (one with water pressure above the ice overburden pressure) lifts
the glacier above otation locally, reducing pressure on the bed beyond the channel. Typical variations are in the 0.1-1 MPa range, safely below the 6 MPa strength of ice. At
night, the channel pressure falls, and the pressure rises on the bed beyond the channel.
One can model the eect of the varying water pressure in the channel by approximating it as a Hertzian indentor|a region in which pressure is raised. The analytic
solution for stresses induced by the Hertzian indentor is given by Jaeger and Cook (1979)
as
14
yy =
P 1 2 sin(1 2) cos(1 2)
P xx = 1 2 + sin(1 2) cos(1 2)
P xy = sin(1 2) sin(1 + 2)
;
;
;
;
;
;
;
;
(3.3)
(3.4)
(3.5)
where xx is the normal stress parallel to the x, or horizontal direction yy is the
normal stress parallel to the y, or vertical direction xy is the shear stress P is the
applied pressure and 1 and 2 are given by Figure (3.1).
3.3 Model Summary
The daily summer periodicities in water delivery to the glacier bed cause variations
in the local stress regime. These stresses provide the necessary initial values for oscillations in the stress regime. The changes in the stress regime cause the changes in stress
intensity factor, KIc , given in equation (3.1). The changes in the stress intensity cause
fatigue in the bedrock as given by equation (3.2). Fatigue is the fracture-propagating
mechanism for the model. However, in order for the fractures to propagate horizontally,
the maximum stress must be in the horizontal direction. Therefore, the channel stress
must be linearly superposed on a compressive, horizontal stress in the bedrock.
15
Fig. 3.1. The geometry of a Hertzian indentor with width W. (Modied from Jaeger
and Cook, 1979)
16
Chapter 4
Methods and Results
4.1 Crack Propagation Vectors
Cracks propagate in the direction of the greatest principal compressive stress,
1 (Atkinson, 1987). In order for a crack to propagate horizontally in the near-surface
below a glacier, the greatest principal stress must be horizontal. However, the subglacial
channel will produce a nearly vertical fracture propagation direction, especially in the
high stress areas. For example, Figure 4.1 illustrates the crack-propagation directions
around a channel modeled as a Hertzian indentor and it shows that the crack propagation
directions will be near-vertical. Therefore, the channel stress must be superposed on
a near-surface compressive stress. Near-surface stresses arise from tectonic, erosional,
or other causes, and are observed to have the greatsest compressive stress, 1 , nearly
horizontal (e.g. Engelder and Sbar, 1984 Plumb et al., 1984a Plumb et al., 1984b
Engelder and Geiser, 1984 Holzhausen, 1989).
The stress 1 will be more horizontal than vertical when the eective pressure
from the channel is less than the regional near-surface compressive stress. Indeed, Figure
4.2 illustrates the rotation of the vectors from Figure 4.1. Because the vectors in Figure
4.2 are closer to horizontal near the center of the stress eld, one can assume that these
fractures will be the horizontal fractures needed by Hallet's model.
17
Fig. 4.1. The crack propagation vectors resulting from an overpressured channel on
bedrock without a near-surface stress. The channel is 2 m wide and Pe is 1.0 MPa.
18
Fig. 4.2. The crack propagation vectors resulting from an overpressured channel on
bedrock with a near-surface stress. The near surface stress is 10 MPa. The channel is 2
m wide and Pe is 1.0 MPa.
19
4.2 Fatigue Modelling
Equation (3.2) can be solved for the change in crack length from fatigue:
da = C (KI )m dN
(4.1)
This equation then yields the iterative solution for the crack length:
ai = ai;1 + da
= ai;1 + C (KI )m dN
(4.2)
Equation (4.2) depends upon the a0 value and the change in stress intensity factor. The
stress intensity factor depends upon the crack length and the least principal stress, 2 .
Because the crack length is constant for an individual calculation, the change in stress
intensity factor only depends on the change in 2 . The change in 2 is the dierence
between the daytime and nighttime extremes in Pe .
The calculated change in stress depends on the channel size, the eective pressure,
and distance from the channel. The depth to which a channel aects signicantly stresses
scales with the width of the channel, and is approximately the width of the channel. This
relationship is illustrated by the dierences between Figures 4.2 and 4.3.
The constant background, near-surface stress must also be accounted for in the
computations. Fundamentally, if the background stress exceeds the strength of the rock,
then the rock fails regardless of the number of cycles. A numerical constraint must
be introduced to quantify this condition. The constraint of equation (2.6) claries the
20
Fig. 4.3. The crack propagation vectors resulting from a larger overpressured channel
on bedrock with a near-surface stress. The near-surface stress is 10 MPa. The channel is
5 m wide and Pe is 1.0 MPa.
21
relationship between the background stress and crack growth.
4.2.1 Model Code and Parameterizations
The code to model the eects of fatigue was written in Matlab. Several assumptions went into writing the code. First, only the stress regime directly below the center
of the channel was modelled, because this location provides the highest likelihood of
fracture generation. In addition, initial aws were assumed to be 2 cm in width and
oriented parallel to 1 . The length of 2 cm corresponds well with the work of Segall
(1984). Finally, C is assumed to be 5 10;10 m1=2 MPa;1 , which is the mean value of
C from Kim and Mubeen (1981). The parameter m is assumed to be 11.8, because this
value corresponds to the sample that endured the greatest number of cycles in Kim and
Mubeen's experiments.
The four free variables in the experiment are KIc , depth below the channel, Pe ,
and the magnitude of 1 . Additionally, a critical crack length is assigned so that the
model can compute the number of cycles necessary to attain such a crack. The critical
crack length corresponds to a clast whose erosion is rate-limited by the horizontal fracture
process.
Table 4.1 gives the ranges for each variable and the number of values of each
variable for which the model was solved. The model assumes that 2 = 0:561 based
upon the data presented by Holzhausen (1989).
22
Variable
KIc
1
Pe
Depth
Critical
Crack
length
Table 4.1. Model Parameters for Each of the Four Variables
Maximum
Minimum
Number Reference for ValValue
Value
of Values ues
1=2
1=2
3 MPa m
0.2 MPa m
5
(Atkinson, 1984)
12 MPa
0 MPa
5
(Plumb, et al.,
1984a, 1984b)
6 MPa
0.25 MPa
4
(Hallet, 1996)
1m
0.01 m
10
Depth to which
the stress regime
has a large eect
2m
0.03 m
10
Clast Sizes in
Glacial Sediments
4.3 Results
The results fell into three primary categories: i) the horizontal fractures propagated without fatigue ii) fatigue drove the fracturing until the crack became long
enough to dominate KI and iii) fatigue could not activate the aws to drive them to a
reasonable clast size before 5,000,000 cycles had been completed. (Glaciers have approximately 100 days per year that water is delivered to the bed, making those 5,000,000
cycles correspond to 50,000 years of fatigue.) Figure 4.4 delineates these three zones
rather well. In the gure, the blue area (upper left portion) represents the conditions for
which the near-surface stress propagates cracks. The red area (lower right portion) of
the graph represents a high fracture toughness, KIc , or a low stress, such that no fracturing occurred in 5 million cycles. The rainbow-like area of the graph represents those
background stresses and fracture toughnesses that need fatigue to drive the horizontal
fractures before the crack becomes large enough to \propagate itself".
23
6
x 10
7
4
Near-surface Least pricipal stress (sigma 2) (MPa)
6
3.5
3
5
2.5
4
2
3
1.5
2
1
1
0
0.5
0
0.5
1
1.5
2
2.5
3
Cycles to failure
Critical Stress Intensity Factor (MPa m 1/2)
Fig. 4.4. The relationship of the background stress to the critical stress intensity factor
via the number of cycles to fatigue. The number of cycles as a function of background
stress and KIc value. Other parameters are: the critical fracture length is 0.46 m, the
depth is 0.23 m, the channel width is 2 m, and Pe is 6 MPa
24
To gain a better appreciation for these three categories, a depth{clast plot was
made. Figure 4.5 lies in the transition area of Figure 4.4 where fatigue helps propagate
the crack. Figure 4.5 illustrates that the timing for fracture is similar for a given critical
fracture length for depths >10 cm below the surface. From 10 cm depth, rates of fracture
propagation increase toward the base of the ice. This change in the fracture propagation
rate is a result of the exponential nature of equation (3.2).
25
Fig. 4.5. Cycles to failure as a function of depth in the bedrock and critical crack, or
clast, size. Other parameters are: Pe is 6 MPa, 2 is 1.68 MPa, KIc is 1.6 MPa m1=2,
and the channel width is 2 m.
26
Chapter 5
Discussion and Conclusions
Glaciers erode through subglacial stream processes, by plucking clasts from their
beds, and by abrading or \sandpapering" their beds using plucked clasts, or clasts from
other sources (e.g. Sugden and John, 1976). The importance of direct stream erosion
is debated, but the widespread occurrence of features from plucking and abrasion on
deglaciated rock argues for their importance (Sugden and John, 1976). Of these, the
observation that glaciers commonly discharge clasts of their bedrock that have not been
\worn out" by abrasion argues for the common dominance by plucking.
Previous studies of plucking have assumed the existence of horizontal fractures
or bedding planes and have modelled vertical fracture propagation. Here, the situation is modelled where horizontal bedding planes are absent, and horizontal fracture
propagation is required for plucking. From the results, one nds three regimes: meanstress-dominated, fatigue-dominated, and abrasion-dominated.
In the mean-stress-dominated regime, non-glacial, surcial stresses are large
enough to produce fractures more quickly than glacial processes. Vertical joint propagation is likely to be the rate-limiting step in plucking, and models such as that of
Hallet (1996) are appropriate.
In the abrasion-dominated regime, high rock strengths or low stresses produce a
situation in which horizontal fractures are not produced at a signicant rate. For a rock
27
mass initially lacking abundant horizontal fractures, plucking will be slow, and abrasion
by pre-existing clasts or clasts supplied to the glacier bed from ice-marginal regions likely
will be dominant.
Between these extremes lies the fatigue-dominated regime, in which variations in
subglacial hydrology can drive horizontal fractures. This regime is favored by intermediate non-glacial stresses and rock strengths, and by variable glacier water pressures forced
by abundant meltwater supply. Relative rates of horizontal and vertical crack growth will
determine which is rate-limiting in most cases, horizontal crack growth is likely to limit
plucking.
If horizontal crack growth is rate-limiting, then the erosion rate will depend on the
number of cycles required for failure, Nf . The depth scale to which basal channels have
signicant eect on the subglacial stress regime scales with the channel width, W . The
number of cycles per year, nc is controlled by meteorological conditions, but is roughly
100. Assuming that channels are su ciently close together that all of the glacier bed is
aected by their pressure uctuations, that the depth to which the channels aect the
bedrock is a fraction of the channel width, , and that horizontal fracture propagation
is rate-limiting, the erosion rate, r_ , will be approximately
wn
r_ = N c
f
(5.1)
If the water-pressure uctuations aect only a fraction, , of the bed at a time but
c
migrate over time, then r_ = wn
Nf .
Results such as those of Murray and Clarke (1995) suggest that W is on the
28
order of 1{10 m. Similarly, nc is known within an order of magnitude in most cases. The
number of cycles to failure, Nf , is a very strong function of water-pressure variability
and mean stress. Because water-pressure variability increases with water supply (Alley
et al., in press), one expects erosion to be rapid well downglacier where water ux is
high. Erosion also should be rapid in regions of high tectonic stresses and weak rocks.
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29
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