5.1. Relation Between Normal Stress, Porosity, and Wear Rate From Experiments
 Figure 8 shows log wear rate (m m−1) plotted against lognormal stress for each of the rock types studied. For each sample except Penrith sandstone, the wear rate is very sensitive to changes in normal stress, and on the log/log plot the trends appear to be approximately linear. The Penrith sandstone data may represent a tendency to a more linear wear rate versus stress relationship at lower contact stresses and/or because it is a higher-porosity rock. The plot shows that the more porous rocks display higher wear rates, even at lower normal stress levels. The overall form of the data for the entire suite of rocks suggests that it may be reasonably represented (for a single clast) by a wear law of the form
where σ is effective normal stress (MPa); A, n, and m are empirical constants; dT/dx is a dimensionless wear rate (meter of wear per meter of shear displacement); and ϕ is porosity (vol voids/(vol voids + solids)). Figure 8 also shows fits to this equation by multiple linear regression after taking logarithms, which yielded the parameters, A = 10−8.11, n = 8.33, and m = 4.50, with a standard error (in log dT/dx) of ±0.72. Except for the trend for the Penrith sandstone data, this fit describes the data fairly well, and provides a provisional basis for the estimation of subglacial abrasion rates for different rock types provided the porosity is known.
Figure 8. Plot of log wear rate versus lognormal stress for all samples tested. The straight lines show a multiple linear regression fit to the data using the equation log dT/dx = log A + n log σ + m log ϕ, with normal stress in MPa and wear rate in m m−1, to yield the parameters shown.
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 For all the rock types studied, the maximum applied stress is substantially less than the unconfined compressive strength of the rock (Table 1), with the exception of Penrith sandstone, which was subjected to contact stresses up to ∼50% greater than the unconfined strength. For contact loading between two cylindrical surfaces, a high mean stress (hydrostatic part of the total stress tensor) is induced beneath the loaded surfaces [Hills et al., 1993]. It seems likely that this confinement effect suppresses total crushing failure in the same way that brittle failure is avoided in indentation hardness testing of a range of materials that would normally be considered to be brittle.
 Archard  proposed a model for the wear process in which the wear rate is linearly proportional to the applied normal stress. This arises from the assumption that wear rate is proportional to real contact area between the surfaces and that this is in turn linearly proportional to the applied load. This has been verified empirically for a number of materials, both ductile and brittle [e.g., Costa et al., 1997], and a number of subsequent theoretical studies have supported this conclusion (review by Wang and Hsu ). For brittle solids, the generation of wear debris depends on the propagation of tensile indentation cracks nucleated around the periphery of the indenter [Lawn, 1975], the lengths of which (and hence the amount of debris produced) are expected to be approximately proportional to applied load in the absence of any kinetic effects attributable to environmental fluids and thermal fluctuations. Wang and Scholz  and Yoshioka  found that a linear wear rate versus normal stress relationship applies to fault surfaces in a ring shear apparatus (in which wear debris is allowed to accumulate with displacement). Nevertheless, many studies have reported nonlinear (power law) relationships between dry wear rate and normal stress. For example, Conway and Pangborn  obtained normal stress exponents up to 1.74 for brittle abrasive wear of SiAlON ceramic against steel, and Kong et al.  a normal stress exponent of 3 for mullite against zirconia-toughened mullite (ZTM) under lubrication by machine oil. Moore and King  reported power law wear with a stress exponent ranging between 1 and 1.6 under normal stresses up to 1.5 MPa for sintered silica worn on silicon carbide. Even for metal-on-metal sliding wear can be nonlinear. Tu et al.  reported strongly accelerating wear rates with normal stress in copper sliding against carbon steel.
 Wear rate depends not only on normal stress but also on chemical environment. The presence of an aggressive fluid can substantially enhance wear rates [e.g., Bundschu and Zum Gar, 1991; Dogan and Hawk, 2001; Novak et al., 2001; Frye and Marone, 2002]. Kong et al.  reported substantial enhancement of wear of mullite and other alumina-based ceramics by water, with an hydrolysis reaction implicated in the wear process (tribochemical wear). Additionally, the normal stress exponent increased to more than 4, and some evidence was seen for the existence of a wear mechanism transition to a more nonlinear behavior with increasing normal stress.
 The extreme nonlinear wear rate as a function of normal stress (stress exponent of 8.3) observed in the present water-saturated experiments may be explained by subcritical (e.g., stress corrosion) crack growth, in which crack growth rates depend on the kinetics of hydrolysis of stretched silicon-oxygen bonds at crack tips. For quartz in this regime, crack velocity v (m s−1) varies with mode I stress intensity factor KI (MPa m1/2) as [Atkinson, 1984]
Thus crack growth rates would become enhanced relative to dry conditions, producing wear debris more efficiently and a stress exponent of wear rate similar to the KI exponent in equation (6), until crack growth rates become so high that water vapor is unable to diffuse to crack tips, leading to hardening. Figure 9 shows schematically the expected behavior, which is similar to that reported by Kong et al.  for alumina-ceramic materials. This argument also implies that at a given normal stress, slow sliding will produce more wear debris than fast sliding. All of our present experiments were performed at a constant sliding rate. For the crystalline rocks this was on the order of 600 m yr−1, or 50 times faster than the sliding rate of the Findeln Glacier. From our present data we cannot demonstrate any specific time-dependent effects, but for slower sliding we would expect some reduction of the normal stress associated with a given wear rate.
Figure 9. Schematic diagram showing subcritical crack growth may accentuate overall wear rate and make the wear versus stress relationship nonlinear.
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 Only monolithologic erosion was studied in these experiments. It might be anticipated that if dissimilar wheels were used, the rate of wear of the weaker rock type would be greater than that of the stronger, and also that the rate of wear of the weaker rock might be faster than for wear between wheels of the same (weaker) rock type. Such a speculation would require further experiments for evaluation.
5.2. Temperature Effects
 The experiments reported here were all performed at room temperature, some 20°C above the temperatures encountered at the base of a glacier. Such a small temperature difference would not be expected to have any detectable effect on the kinetics of the wear process. Brittle deformation processes are generally very insensitive even to larger temperature changes. The only additional factor at work at the base of a glacier is the possibility of freeze-thaw effects on a diurnal basis, enhancing the kinetics of crack growth. We suspect this is insignificant for enhancing the growth of microcracks on the order of a few microns or tens of microns in length that would be important for producing clasts of silt or finer size. Freeze-thaw effects should only be important for the enhancement of the growth rate of much longer cracks (several cm) so that greater stress intensification at the crack tip can be produced through the leverage effect. The latter effect could, therefore, contribute to the plucking of rock fragments from the glacier base, as it does for the fracturing of rocks under subaerial conditions.
5.3. Comparison of Experimental Data With Observations of Natural Subglacial Abrasion Rates and Tectonic Fault Abrasion Rates
 As pointed out earlier, natural abrasion rates can be estimated from the rate of production of the suspended sediment load in the outflow of subglacial streams. The figure thus obtained for the Findeln Glacier during the summer period (Figure 10) is ∼0.5 mm yr−1, assuming a uniform rate of abrasion over the whole of the present area of the glacier [Lee, 2001]. This figure is very much in the middle of the range found for temperate glaciers [Hallett et al., 1996]. The average displacement rate of the surface of the glacier is ∼10 m yr−1 [Lee, 2001]; hence the average wear rate is ∼0.05 mm m−1, or 5 × 10−5 m m−1, assuming no ice flow past the entrained clasts. Observation of the basal ice of the Findeln Glacier at the snout and at the sides shows it to be very clean, with entrained debris certainly less than 10% volume. Thus to account for the inferred average abrasion rate, individual clasts must therefore have their dimensions reduced at least ×10 more rapidly. This range of wear rates (m m−1) corresponds to the range of wear rates that were measured in our experiments (Figure 8).
Figure 10. Cumulative wear rate estimated from suspended sediment flux in the outflow from the Findeln Glacier. Each datum point represents wear and glacier surface displacement corresponding to a measurement period of a few weeks during the summer of the year shown, accumulated year-on-year to yield the wear rate, but does not record the total displacement and sediment production for the whole of the calendar year indicated.
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 Estimates of the rate of production (meters gouge per meter displacement) of tectonic fault gouge can be made from experiments and field studies. Experimental data [Yoshioka, 1986; Scholz, 1987; Power et al., 1988] showed dT/dx to be about 2 × 10−4 for granite and 1 × 10−3 for sandstone under a normal stress of 20 MPa, and increasing with normal stress. These rates are ∼20 times faster than observed for comparable rock types and normal stresses in our study. The wear rate deduced from studies of natural fault zones is even greater, lying between 10−1 and 10−3 [Scholz, 1987], 20 to 2000 times greater than subglacial wear rates.
 There are important differences between subglacial abrasion beneath temperate glaciers and tectonic fault wear. In the former, wear products are ultimately largely washed away by subglacial streams (but not the case in cold-based glaciers), and the ice conforms as it moves by plastic flow to the shape of the glacier bed. In tectonic faults, wear debris accumulates, and in most cases forms at depths in the Earth under combinations of normal and shear stress that are much greater than encountered in the subglacial environment, leading to greater damage to the wall rock. If faults were perfectly planar, slip would eventually be focused within the layer of gouge, but their natural roughness on a wide range of length scales means that the damage zone must widen with displacements up to more than 10 km as progressively longer-wavelength asperities are worn down. The inability of the wall rocks to conform geometrically without large stress concentrations developing must lead to rapidly widening comminution of the wall rocks. Thus wear in tectonic faulting is not directly comparable with subglacial abrasion.
5.4. What is the Effective Rock-on-Rock Normal Stress Required for Abrasion?
 If a glacier has a local thickness z, and is partially saturated with water, on a slope of dip angle β, the local effective normal stress is
where ρ is the ice density, ρw is the water density, g is the gravitational acceleration, and f is the fraction of the ice thickness that is saturated with water. Taking the Findeln Glacier as an example, the fraction f varies substantially on a diurnal basis, from ∼0.65 during daytime in summer to ∼0.35 at night, with corresponding out-of-phase daily fluctuations in the discharge of the subglacial streams (Figure 11). For an ice thickness of 160 m (in its central region) and an average basal slope of 3.6°, this leads to an effective spatially averaged normal stress varying from about 1.0 to 0.4 MPa. Assuming a basal debris concentration of less than 10%, a wear rate dT/dx of more than 0.5 mm m−1 shear displacement is required to account for the observed suspended sediment flux. According to the experimental wear data (Figure 8) for the rock types beneath the Findeln Glacier, sliding under an effective normal stress between particles and bed of ∼30 MPa is required to generate this flux (bear in mind that this figure might be less at lower time rates of sliding). Yet the experimental data imply that under an effective normal stress of only 1 MPa the wear rate would be virtually zero.
Figure 11. Diurnal fluctuations in stream discharge (m3 s−1) from the snout and water level (m above glacier base) in borehole 99.33, situated ∼1.2 km from the snout of the Findeln Glacier in the summer of 1999 (unpublished data of A. Lee, N. Rutter, and D. Collins). The two types of data vary in sympathy but with a few hours phase shift.
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 Experimental data show that lithic fragments entrained in ice in temperate glaciers actually experience much higher normal stresses than that due simply to the weight of overlying ice. In a seminal series of experiments, Iverson  showed that normal force Fn between a spherical clast embedded in ice and a rock bed was enhanced in proportion to the downward velocity Vn of the ice as it pressure-melted against its bed (Figure 12). The enhanced force was due to the “frictional” drag of the ice against the clast, as predicted by Hallett  and Shoemaker . Thus Fn = ΨVn, where the drag coefficient Ψ is given by
where r is the radius of the clast, η is the effective viscosity of the ice, and r* is the clast radius at which the transition from regulation flow to plastic flow around the clast occurs [Hooke, 1998]. At large clast sizes (r > 10 cm approximately) plastic flow dominates. Shear stress causing sliding was also observed to increase in correspondence with an increase in bed-normal ice velocity. Iverson  measured normal forces in excess of 500 N on a sphere of 27 mm radius under a downward ice velocity of 200 mm yr−1 with an ice pressure of 1 MPa. For the above spherical clast the local normal stress is ∼5 MPa after wear has increased the contact area to 1 cm2, and higher still at smaller contact areas. Smaller diameter clasts are expected to produce higher local pressures, and higher pressures are to be expected along stoss surfaces of basal irregularities. Iverson  also noted that measured forces tended to exceed those calculated from theory. Because pore water pressure must be subtracted from this enhanced normal stress, the effect of pore pressure becomes less than it might otherwise have been. On the other hand, some enhancement of local pore pressure is to be expected as the water produced by pressure melting will be transiently at the local ice-rock normal stress before it has leaked away by permeation.
 High local normal stresses beneath clasts are also implied by in situ measurements of spatially averaged shear stress at the Svartisen Subglacial Laboratory, northern Norway [Iverson et al., 2003], in ice with 2 to 11% volume of debris. Spatially averaged shear stress levels up to 300 kPa were measured beneath 210 m of ice sliding at 0.1 to 0.2 m d−1. Assuming the shear load is taken on 5% volume of debris particles, and none by the water-lubricated ice/rock contact, local shear stresses of ∼6 MPa are implied, or a local normal stress beneath particles of ∼10 MPa, assuming a friction coefficient of 0.6 for rock-on-rock sliding. The implied tenfold enhancement of effective normal stress beneath clasts was attributed to stress enhancement due to pressure-melting against the bed. The authors point out that there is a great deal of uncertainty in the estimation of bed/clast stresses arising through this mechanism. Nevertheless, it seems likely that in this way our experimental results can be reconciled with the observed high rates of production of abrasive wear debris beneath temperature glaciers, such as the Findeln Glacier.
5.5. Relative Importance of Sliding Rate Versus Normal Stress in Abrasion
 Our experimental data demonstrate how the production rate of wear debris per meter of clast displacement is very sensitive to normal stress between clasts and the glacier bed. The displacement rates measured on the surface of a glacier can be decomposed into motion due to internal distortion of the whole glacier through ice flow and the rigid body sliding of the glacier over its bed, the latter component giving rise to abrasion as long as clasts are carried along with the ice. The basal sliding rate appears to be more important than the internal distortion contribution to measured surface displacement. Jansson  decomposed these two components for the Findeln Glacier, showing that basal sliding contributes ∼80% of the total displacement, with the sliding rate dx/dt depending on the spatially averaged effective normal stress at the glacier base due to the weight of the overlying ice according to
where C = 96 when sliding rate is in m yr−1 and effective normal stress is in MPa.
 The relative importance of effective normal stress and sliding velocity for the production of abrasive wear debris can be made by combining equations (5) and (9) to eliminate displacement. This gives for the rate of wear per unit time
where D incorporates A, C, and the porosity term. The remaining strong dependence of wear rate dT/dt on effective stress implies that most wear is produced when the normal stress is high (and water level low) despite the slowing down of the sliding rate as the water level falls but provided that local friction between clasts and bed does not cause them to stop sliding. On the other hand, if the local clast/bed normal stress is determined mainly by the rate of melting against the bed, this may conversely increase daytime wear rates if basal melting is enhanced by the flux of meltwater through the glacier.