D

Figure 7.18. (a) A grain bridge, formed by nearly coaxial alignment of several grains in a deforming granular medium, may fail by: (b) fracture of a grain;or (c) slip between grains. (d) Stresses at contacts between grains are reduced when additional particles occupy pore space. Heavy arrows show the shear stress applied to material, t, and the component of this stress along a grain bridge, a. (Modified from Hooke and Iverson, 1995, Figure 1.)

Figure 7.18. (a) A grain bridge, formed by nearly coaxial alignment of several grains in a deforming granular medium, may fail by: (b) fracture of a grain;or (c) slip between grains. (d) Stresses at contacts between grains are reduced when additional particles occupy pore space. Heavy arrows show the shear stress applied to material, t, and the component of this stress along a grain bridge, a. (Modified from Hooke and Iverson, 1995, Figure 1.)

Figure 7.19. Grain bridges in a two-dimensional array of photoelastic disks under shear. (Adapted from Howell et al., 1999.)

along a three-dimensional array of routes, forming what could be called a grain-bridge network (Iverson et al., 1996). The complexity of a two-dimensional network can be studied by shearing an array of photoelastic disks and viewing them in transmitted polarized light (Figure 7.19). Changes in the optical properties of disks under stress makes them appear lighter. Thus, the bridges show up as chains of lighter disks in Figure 7.19.

For deformation to occur, grain bridges must fail. Failure may be a consequence either of fracture of a grain (Figure 7.18b) or of slippage between grains (Figure 7.18c). Fracture is most likely when two adjacent grains are of roughly equal size and when the space between them is not filled with smaller grains that absorb some of the stress. Slip between grains occurs when resolved stresses parallel to contacts between particles are greater than tp. Because the deviatoric stress required to fracture a grain varies with particle size, and because contacts between grains may have different orientations leading to different resolved stresses, there must be a wide range of bridge strengths.

Figure 7.20. Grain-size distributions in two subglacial tills that were deforming. Fractal dimension of each till is shown. (Modified from Hooke and Iverson, 1995, Figure 2.)

10 1012 1011 1010 109 108

106 105 104 103 102 101 10° 10-1

\

• Storglaciàren m = 2.91 ~

▲ Engabreen m = 2.89

_

\# -

_

_

A\\

-

XA*

-

a\7\ -

-

-

• •••1

0.001 0.01 0.1 1 Particle size, mm

Grain fracture alters the granulometry of a material. Biegel et al. (1989) argue that the end product of this process is a granulometry that maximizes the support that each particle receives, and thus minimizes stress concentrations capable of causing fracture. For example, forces between particles in Figure 7.18d are distributed over several contact points, so local stresses are less likely to reach a level that will cause fracture. The granulometry that provides maximum support, according to Biegel et al., is one in which no two particles of the same size are in contact. Such a material has a fractal particle-size distribution with a fractal dimension of ~2.6. That is, if No is the number of particles of a reference size, do, then the number of particles of size d, N(d), is:

The fractal dimension is m. (As is evident from Equation (7.19), fractal size distributions appear to be the same at all scales. Thus, if there is one particle of unit size in a field of view, there will be 10m particles that are 1/10 this size, regardless of the units used in making the measurement.) Sammis et al. (1987) have shown that gouge from the Lopez Canyon Fault in California has such a particle-size distribution.

Deforming subglacial tills also have a fractal granulometry, with a fractal dimension close to 2.9 (Figure 7.20). This suggests that grain fracture may play an important role in till deformation. That the fractal

Table 7.2. Sliding speed, till thickness, and strain rate in the till beneath various glaciers

Sliding speed, Till thickness,

Glacier

ub, m a-1 ht, m

et = ubht, a 1

Reference

Blue Glaciera

4 0.1

40

Engelhardt et al. (1978)

Breidamerkurj okull

24 0.5

48

Boulton and Hindmarsh

(1987)

Storglaciaren"

10 0.2

50

Hooke etal. (1992)

Trapridge Glacier

33 0.5

66

Fischer and Clarke (1994)

Whillans Ice Stream

450 6b

75

Alley et al. (1987a)

Whillans Ice Stream

360 2.5-0.5

144-720

Tulaczyk et al. (2001b)

Whillans Ice Stream

360 0.3-0.03

1200-12000

Engelhardt and Kamb

(1998)

Columbia Glacier

1300 0.65

2000

Humphrey etal. (1993)

a Values of et for Blue Glacier and Storglaciaren may be maximum estimates of the critical strain rate as the deformation is inferred to extend to the till/bedrock interface.

b Thickness of deforming till layer beneath Ice Stream B is inferred on the basis of geophysical data, whereas other values of ht are based on more direct measurements.

a Values of et for Blue Glacier and Storglaciaren may be maximum estimates of the critical strain rate as the deformation is inferred to extend to the till/bedrock interface.

b Thickness of deforming till layer beneath Ice Stream B is inferred on the basis of geophysical data, whereas other values of ht are based on more direct measurements.

dimension is larger than the ideal of 2.6 is attributed to the production of fine material by abrasion, a process that would be inhibited by the higher effective normal pressures characteristic of deformation in active faults (Hooke and Iverson, 1995). However, more work is needed fully to understand the processes that give rise to fractal size distributions.

Strain rates in subglacial till

Estimates of strain rates in till based on field data all exceed 40 a-1 (Table 7.2). Many of these may be minimum estimates as observations, discussed at greater length below, suggest that deformation is commonly concentrated in shear zones. The thickness of these shear zones may be limited by the length of grain-bridge networks. The length of a network may be limited if stress build up is slow and multiple small adjustments between particles relieve stresses. Thus, thicker shear zones may indicate higher rates of deformation.

Till rheology - the Coulomb plastic model

Our discussion thus far has focused on the strength of granular materials. There is a considerable volume of literature on this topic because of the interest in the conditions under which slopes fail, leading, for example, to landslides or the collapse of highway embankments. In contrast, studies of the time-dependent behavior of deforming granular materials are less common. Furthermore, they often deal with materials containing an abundance of clay minerals, as clays are an important constituent of many slow-moving landslides.

As noted, failure occurs when the failure strength of a material is exceeded (Figure 7.15). We, however, are interested in the rate of steady deformation some time after failure. Thus, the relevant measure of strength for studies of rheology is the residual strength. More specifically, we need to know whether the residual strength increases with strain rate, other factors such as effective pressure, granulometry, mineralogy, and so forth, remaining constant (Kamb, 1991). If such an increase occurs, the strain rate may be a unique function of the applied stress, and a "flow law" for till may exist. If the residual strength does not increase with strain rate, till is a perfectly plastic substance; once the residual strength is reached, it will deform at whatever rate is necessary to prevent the applied stress from rising above that strength.

In Chapter 4 we discussed possible mechanisms that might control the rate of deformation of ice. Let us now do the same for till. The principal processes we have discussed are dilation and failure of grain bridges. Dilation occurs early in the deformation process, and once the medium is dilated it remains so. Thus, dilation should not be rate controlling, and in the absence of repeated formation of grain bridges, we might expect the material to deform steadily and homogeneously, once the failure strength is exceeded. Grain bridges do form, however, and deformation proceeds only when a bridge fails. This suggests that failure of grain bridges may be the rate-controlling process in till deformation. If this is the case, and if the formation of grain bridges is stochastic in time and space, then a mechanistic rheological model for till deformation should be based on these processes. Analysis should focus on the frequency of failure of grain bridges and on the amount of deformation resulting from each failure.

Studies of processes that are thermally activated, such as the creep of ice (Equation (4.6)), provide a conceptual framework for such a model. In thermally activated processes, the process operates or proceeds when a certain energy barrier is exceeded. In the creep of ice, the barrier is the energy needed to break an atomic bond, thus allowing movement of a kink in a dislocation (Figure 4.6). Fundamental to the theory of thermally activated processes is a premise, based on principles of statistical mechanics, that the probability distribution, p(f), of energy levels, f in atomic bonds is given by:

where A and a are constants (Glasstone etal., 1941,p. 159). In containers filled with beads and subjected to a normal load, the distribution of force levels at intergranular contacts is, indeed, given by Equation (7.20) with f now defined as the force at such contacts (Liu et al, 1995). Thus, it seems plausible that the theory of thermally activated processes can be adapted to the analysis of deformation of granular materials. Mitchell et al. (1968) and Mitchell (1993, pp. 349-361) have used this approach, and conclude that a relation of the form:

should describe the steady strain rate in a granular material. Here, t is a mean shear stress sufficient to cause deformation and thus maintain dilation, and T and y are constants presumably dependent upon the strength and granulometry of the material.

A flow law for till that is of this form is: t s

To so

(e.g. Mitchell, 1993, Figure 14.15). This relation is consistent with laboratory data. Here, to is normally taken to be the stress at a reference strain rate, so; to must be greater than s (Equation (7.17)) as the material is deforming. In other words, the Coulomb yield criterion must be met. Thus, for example, if we choose so to be some constant low strain rate, independent of Ne, we can write to = s + As where As is the amount by which to must exceed s to yield that so. The reference stress, to, depends on Ne; an increase in Ne increases to and therefore decreases s as expected.

In engineering tests on sandy materials preconsolidated to 392 kPa, b « 0.013 (Nakase and Kamei, 1986, Figure 14), whereas for materials with significant quantities of clay, it is ~0.043 (Mitchell, 1993, Figure 14.15). Tests on till from beneath Whillans Ice Stream yielded values that ranged from 0.10 for samples preconsolidated at 35 kPa to 0.002 for samples preconsolidated at 343 kPa (Figure 7.21) (S. Tulaczyk, written communication, 8/03). Nakase and Kamei's value is broadly consistent with the latter (Figure 7.21). Tulaczyk et al. (2001a) found that the preconsolidation stresses of samples from beneath Whillans Ice Stream were typically 10-20 kPa, soavalueof b of ~0.1 is probably appropriate, at least for these fine-grained tills.

Treating so as a constant, Equation (7.22) may be written:

where ki = eoe-1/b. Comparing this with Equation (7.21) we see that ki = T and 1/bTo = y . Based on the experimental values of b, it is clear

Figure 7.21. Effect of preconsolidation stress on b. Data are based on undrained triaxial tests on samples of till from beneath Whillans Ice Stream (Tulaczyk et al., 2000a). Values of b were kindly calculated by Tulaczyk (written communication, 8/03). Error bars are estimated from plots supplied by Tulaczyk. Line is drawn by eye, ignoring points with largest uncertainties. Point labeled N&K is from Nakase and Kamei (1986).

Preconsolidation stress, P0, kPa

Figure 7.21. Effect of preconsolidation stress on b. Data are based on undrained triaxial tests on samples of till from beneath Whillans Ice Stream (Tulaczyk et al., 2000a). Values of b were kindly calculated by Tulaczyk (written communication, 8/03). Error bars are estimated from plots supplied by Tulaczyk. Line is drawn by eye, ignoring points with largest uncertainties. Point labeled N&K is from Nakase and Kamei (1986).

that strain rates should increase substantially with only a small increase in t , as Kamb (1991) recognized.

Such a dependence of ¿on t in till is also consistent with field measurements on Storglaciaren (Hooke etal., 1997). The measurements were made by inserting instruments into till beneath ~ 120 m of ice, using boreholes through the glacier to gain access to the till. One instrument consisted of a cylinder with conical ends (the "fish") that was dragged through the till by a wire connected to a load cell. The force required to drag the fish varied between about 407.4 N and 408.6 N during a period of several days when the speed of the glacier varied diurnally (Figure 7.22). The variations were basically in phase with those in Ne, which is consistent with Equation (7.17). However, the force was not related to the surface speed. As the speed with which the fish was pulled through the till should have varied in phase with the surface speed and the

0.20

03 CL

0.00

N Surface N\ speed

0.04

N Surface N\ speed

0.03

0.02

0.00

0.03

0.02

0.00

Time, days

Figure 7.22. Relation among drag on a cylinder pulled through subglacial till beneath about 120 m of ice, surface speed, water pressure in a nearby borehole, and shear strain rate in the till. Data are from a period of about 10 d in August, 1992, at a time when all parameters were varying diurnally, and are "stacked" by averaging values obtained at the same time of day each day. (After Hooke et al., 1997. Reproduced with permission of the International Glaciological Society.)

variations should have been of similar amplitude, it appears that the force on the fish did not depend significantly on its speed through the till, and hence did not depend on the strain rate in the till immediately adjacent to it. This is consistent with a constitutive relation of the form of Equation (7.23).

If till is a Coulomb-plastic material (Equations (7.17) and (7.23)) one would also expect that increases in Ne would strengthen the material and thus decrease the strain rate, and conversely. This may be seen in many of the small diurnal variations shown in Figure 7.23 (see especially July 28-August 1 and August 7-10). (Because changes in pressure in a borehole take time to penetrate into the subglacial till, s lags Ne slightly in these diurnal signals.)

The alert reader may have noticed that the variations in Ne and s in Figure 7.22 are in phase, contrary to what one would expect in a Coulomb-plastic material. The reason for this seemingly contradictory behavior becomes apparent if one notes that Ne averages ~0.1 MPa in Figure 7.22, but is >0.4 MPa during the diurnal fluctuations in

Figure 7.23. Relation among shear strain rate in till beneath Storglaciaren, water pressure in a nearby borehole, and surface velocity. (Modified from Iverson et al., 1995.)

Figure 7.23. Relation among shear strain rate in till beneath Storglaciaren, water pressure in a nearby borehole, and surface velocity. (Modified from Iverson et al., 1995.)

Figure 7.23. Low values of Ne correspond to high water pressures, and at high enough water pressures the coupling between the glacier and the bed may be reduced, thus decreasing the shear stress applied to the till by the ice, and hence £ in the till. The in-phase variations of £ with Ne in Figure 7.22 are attributed to such partial decoupling. The physics of coupling is discussed further below.

Because the strength of granular materials appears to increase slightly with strain rate, it is pseudo-plastic rather than perfectly plastic. However, the rate of increase is so small that referring to such materials as Coulomb-plastic is justified.

Tulaczyk et al. (2000a) have suggested that the increase in strength with increased strain rate is the result of a process called dilatant hardening. Dilatant hardening occurs when a sample dilates but the water content cannot vary; the dilation then reduces the pore pressure, resulting in an increase in Ne, and hence in strength (Equation (7.17)) (Lambe and Whitman, 1969, p. 445). In their tests of till samples from beneath Whillans Ice Stream, Tulaczyk et al. used undrained tests (in which the water content does remain constant) and found that the pore pressure did indeed decrease when the strain rate was increased. They attributed this to dilation and dilatant hardening.

c "co

c "co

Figure 7.24. Experimentally determined relation between stress and strain rate in granular media (solid line) compared with a commonly used viscous constitutive relation for till rheology (dashed line).

Alternatively, one could attribute the increase in strength to the physical nature of frictional processes. As noted above, friction results from interlocking of asperities on two surfaces that are in contact with one another. Failure of a contact between two interlocked asperities may occur by fracture or by dislocation creep within the asperities. In the latter case, a frictional interface that initially appears to be stable may, in time, slip. Thus, if the stress builds up slowly on such a contact, it may eventually fail at a stress that could be sustained, at least briefly, if the stress increased rapidly.

Till rheology - the viscous model

In a number of papers (see, for example, Alley et al., 1987b; Boulton and Hindmarsh, 1987; MacAyeal, 1989), a constitutive relation of the form:

has been used to describe till rheology. Here, k2 is a constant and tc is a critical shear stress below which no deformation occurs. Some authors assume that tc = 0, and others let tc = s (Equation (7.17)). This relation is entirely intuitive; there are no reliable field or laboratory data that support it.

In Equation (7.24), it is normally assumed that 1 < m < 2, so the sensitivity of ¿to t is far less than suggested by Equation (7.23). This is illustrated in Figure 7.24. The other major difference between

Equations (7.23) and (7.24) is in the way in which the effective pressure is incorporated. As to (Equation (7.23)) must be greater than s by an amount sufficient to deform the material at a rate eo, t o must vary directly with Ne, albeit in a poorly constrained and perhaps nonlinear way. Thus, lower effective pressures increase e. In Equation (7.24), lower effective pressures also increase e, but in this case the influence is through both the explicit effect in the denominator and the implicit effect on tc in the numerator.

Viscosity is defined as t/e (Equation (2.17)) implying a linear dependence of strain rate on stress. Equation (7.24) implies a linear (m — 1) or mildly nonlinear (1 < m < 2) dependence of e on t , and thus is generally referred to as the viscous model of till deformation.

Sliding of ice over till

At sufficiently high water pressures, or low effective pressures, a glacier on a deformable bed may become partially decoupled from the underlying till. This may result in a decrease in e despite the decrease inNe (which should weaken the till). As discussed above, this appears to be what happened during the experiment shown in Figure 7.22. Partial decoupling also appears to have occurred during the three periods of low effective pressure on about July 26, August 3, and August 12 in Figure 7.23. Other examples are discussed by Iverson et al. (1995) and Hooke et al. (1989, 1997).

The process of decoupling is rather complicated. As the water pressure rises, it seems likely that a glacier will begin to slide over the till in some places whereas in others, clasts that are gripped in the ice and project down into the till will continue to plough through it, causing local bed deformation. Evidence for such ploughing is occasionally preserved in tills in deglaciated areas (Clark and Hansel, 1989). Because of the extent of the local bed deformation, the stress exerted on a clast by the ice, tc, must be ~5 times the shear strength of the till, s, in order for ploughing to occur. Thus, iff is the fractional area of the interface covered by ploughing clasts, ti is the strength of the interface between the ice and the till, and t ic is the strength contributed by ploughing clasts, we have:

Tic - fckcTc where kc « 5, as noted (Tulaczyk, 1999).

The total strength of the interface must include any traction, tim, between the ice and macroscopically flat parts of the boundary, thus:

Tim — (1 fc^kims where 0 < kim < 1.0. Physically, kim ^ 1as the roughness of the ice-till interface approaches the roughness of the failure planes in the till, and kim ^ 0as the ice-till interface becomes smooth. Thus, in this idealized model the total strength of the interface is (Tulaczyk, 1999):

Two factors that affect kim need to be considered at this point. First, where melting is occurring, there will be a water film between the base of the glacier and the till. The thicker the water film, the lower the value of kim. Secondly, under high effective pressures, ice can regelate into pore spaces in the till (Iverson, 1993). Thus, at high effective pressures the ice base will conform to the till surface better than at low effective pressures, so kim will be higher at high effective pressures. This infiltration of ice into till is inhibited, however, by surface tension between water and sediment grains in capillary spaces. Such surface tension effects are particularly important in fine-grained tills with small pore spaces, so kim will be lower for such tills.

In summary, Equation (7.25) shows that the strength of the interface, ti, will be higher when the till is coarse asfc is then larger and ice will be able to penetrate into pore spaces more readily, increasing kim. The strength will also be higher when the effective pressure is higher and the water film thinner. Tulaczyk has shown that ti is likely to be less than s in fine-grained tills like those underlying Whillans Ice Stream but greater than s in coarse-grained tills like those common to most valley glaciers. Thus, the preferred mode of basal motion is likely to be sliding with ploughing in the former, and coupling with more pervasive till deformation in the latter. Extensive sliding would limit sediment transfer in deforming till sheets, so this again raises the question of how the large volumes of till deposited by the Pleistocene ice sheets were moved. Deformation concentrated in shear zones at depth in the till, discussed below, may provide an answer.

Ploughing

The ploughing process has been studied by Brown et al. (1987). They considered spherical clasts of radius R, embedded half in the ice and half in the till, and suggested that the force required to push such a clast through the till scales with the cross-sectional area of the clast; that is, with R2. As this force must be provided by the ice, and as the ice is at the pressure melting point, regelation and plastic flow must be occurring around the clast. As with obstacles on a glacier bed, the stress that the ice exerts on the clast will be low for both small clasts

Figure 7.25. Stress exerted by ice on a spherical clast half embedded in till. Curves are for ice sliding at 25 and 100 m a-1 and were calculated with the use of Equation (2) of Brown et al. (1987).

Figure 7.25. Stress exerted by ice on a spherical clast half embedded in till. Curves are for ice sliding at 25 and 100 m a-1 and were calculated with the use of Equation (2) of Brown et al. (1987).

1000

0 0

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