Rock-on-rock wear under known normal loads was measured experimentally using 62 mm diameter rotating discs machined from a range of silicic rock types: one granite, two metamorphic rocks, and three sandstones of porosities varying between 7% and 28%. Wear debris was removed continuously from the contacting surfaces by water. We obtained a wear law of the form dT/dx = Aσnϕm, where σ is contact normal stress (MPa), dT/dx is a dimensionless wear rate (meter wear (T) per meter shear displacement (x)), ϕ is porosity, and A, n, and m are empirical constants, 10−8.11, 8.33, and 4.50, respectively. Low-porosity rocks displayed surface polishing, and abraded particle size was substantially smaller than the starting grain size. Whole grains were plucked from high-porosity rocks, without any surface polishing, so that the wear product particle size remained the same as that of the starting material. The empirical wear law describes abrasion beneath a hard-bedded temperate glacier, provided we can estimate both the in situ normal stress between entrained clasts and the bed and the horizontal velocity of the clasts in the flowing ice. Normal stress concentration is expected to develop at clast/bed contacts from drag against ice flowing toward the bed, where pressure melting occurs. These experimental data are consistent with wear rates inferred from silt outflow in subglacial streams, provided a tenfold stress concentration can occur relative to the ice overburden pressure. The rate of production of fine-grained wear products under subglacial conditions is substantially less than by tectonic faulting.
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 Subglacial abrasion is one of the most important processes of upland erosion and is perhaps the principal source of silt-sized particles now preserved in the sedimentary record. Estimates of the rate of abrasion by temperate glaciers can be made from direct measurements of the suspended sediment load of proglacial streams or indirectly from the estimated amount of redeposited sand and silt of glacial origin deposited over a dated time period [Lliboutry, 1994; Hallett et al., 1996]. Reviewing the available data, Hallett et al.  show that erosion rates vary over several orders of magnitudes from less than 0.1 mm yr−1 to more than 10 mm yr−1, but that for hard-bedded temperate glaciers on a range of bedrocks the rate commonly lies in the range 0.1 to 1.0 mm yr−1 over the area of the glacier. The relationships between rates of erosion, sediment production and tectonic uplift in mountain ranges, and the balance between mechanical and chemical degradation have in recent years become of wider interest through the question of possible links (in both directions) between tectonics, erosion and global climate [e.g., Raymo and Ruddiman, 1992; Lamb and Davis, 2003].
 It is generally held that abrasive wear products are not generated by ice sliding over bedrock, but from frictional wear of entrained lithic clasts in the basal ice upon bedrock. Clasts can be incorporated from frost-shattered rocks at the head of the glacier falling into the bergschrund or by plucking (quarrying) by ice at the bed. This implies that clastic debris must be produced at a volumetric rate at least equal to the ultimate production of silt-sized abrasive wear debris, and more so if there is to be some leftover clastic material to accumulate as morainic deposits.
 Intuitively, we might expect an abrasion law to take the form
where dT/dx is the abrasion rate expressed as thickness (T) removed per meter shear displacement (x) of the clasts, Cr is the areal concentration of clasts in contact with the bed, (σ − p) is the effective normal contact stress between clast and bed and p is the pore fluid pressure, S is the total effect of those microstructural properties of the rock that determine its resistance to fracture, and G is an empirical constant. Expressing in the above dimensionless way avoids the immediate need to incorporate sliding rate of the particles on the right-hand side, although ultimately the displacement rate of clasts must be related to the ice velocity. In this formulation we assume that each of the key variables can be separated and that the effective stress law applies. There is a sufficient wealth of data from experimental rock mechanics to show that the latter can be reasonably assumed to be so. Through the Coulomb friction law, we can also anticipate that effective normal stress can be replaced by τ/μ, where τ is shear stress and μ is the coefficient of sliding friction. We cannot assume that f1 and f2 are linear functions. We also cannot assume that the sliding rate of the clasts equals the sliding rate of the basal ice with respect to the bed. If effective normal stress becomes sufficient to inhibit sliding under the shear stress that can be applied by the ice, abrasion stops and viscous or regelation flow of the ice around the clast may occur [Boulton, 1979].
 Surprisingly, there is no comprehensive set of experimental data that describes the rate of rock-on-rock abrasive wear as a function of normal stress. This is essential input into an expression such as equation (1) that might be used to model the abrasive wear process. In this paper we describe a series of experiments to make such measurements, for a range of silicic rock types of different porosities (and hence uniaxial compressive strengths). We then consider the application of such data to the understanding of the subglacial abrasion process, with specific reference to the Findeln Glacier, near Zermatt, Switzerland, from where two of the sample materials were taken. Although our experiments were carried out at room temperature, the insensitivity of rock fracture processes to temperature is such that we would not expect the results to be substantially different at 20°C lower.
2. Experimental Methods
 In the apparatus used, wear was produced at the peripheries of two identical rock wheels, 62 mm in diameter and 17 mm thick, that were forced together under a known load and rotated so that they moved in opposite directions at their points of contact. The wheels were fabricated by slicing diamond cores normal to their lengths. The lower wheel was submersed in a water bath, so that wear products would be flushed away by water. Thus the rock-on-rock contact was maintained with no accumulation of wear products. In this respect the experimental configuration is more like that occurring beneath a glacier, where flowing water removes fine rock debris, than in a tectonic fault zone where wear products accumulate with time.
 The experimental apparatus is illustrated schematically in Figure 1. The lower wheel, driven by an electric motor at 30 rpm, was fixed to the frame of the apparatus. The upper wheel, with its own electric motor driving at 15 rpm, was mounted on a swinging arm so that the wheel centers could approach each other as wear proceeded. Wear was measured as the approach of the two wheels with a linear variable differential transformer (LVDT) having a linear range of ±12 mm. On an extension of the upper swinging arm, weight of 1, 2, or 3 kg could be hung, giving rise to normal forces between the wheels of 19.6, 39.2, and 58.9 N, respectively.
 According to the nature of the rock types used, total reduction in wheel diameter varied from less than 1 mm to more than 5 mm. Experiments under a given load were run for times ranging from ∼1 hour to ∼20 hours. Displacement measurements were logged by computer at an interval of a few seconds. Small or progressive deviations from roundness of the wheels, coupled with the effects of vibration owing to the frictional loading caused a variable degree of scatter in the readings from the LVDT. Readings were therefore averaged over periods of a few tens of seconds to several minutes to eliminate this noise.
 In this study, only pairs of wheels of the same rock type were used. We have not studied wear between dissimilar rock types.
2.2. Study of Recovered Sample Materials
 The wear products were accumulated in a water bath in which the lower wheel was immersed. For each rock type, these were recovered and their particle size distributions were measured using a Malvern Scientifics laser particle sizer. The worn wheels themselves were examined visually and by scanning electron microscopy, to determine the relative importance of damage by abrasion versus fracturing or plucking of individual grains.
3. Rock Types Used and Their Characterization
 Six rock types were used in this study. Two of the rock types were collected from the bedrock over which the Findeln Glacier flows (Figure 2), to compare their behavior with the observed rate of wear in situ as inferred from the flux of suspended sediments in the summer outflow from the glacier. The rocks were respectively a chlorite schist, derived from the metamorphism of original ocean floor rocks that are now preserved in the Zermatt-Saas unit, and a muscovite-quartz schist from the metasedimentary envelope of the Monte Rosa nappe underlying the Zermatt-Saas rocks. Both of these rock types possess a strong schistosity, so the wheels were fabricated with the axis normal to the schistosity. This minimized the tendency for departures from cylindricularity to develop during wear.
Figure 3 shows the particle size distribution of subglacial rock flour produced beneath the glacier, collected near the snout. The particle size distribution is significantly reduced relative to the initial mean grain size of the parent rock. The mineralogy of the rock flour showed it to be derived dominantly from the erosion of the chlorite schist.
 Three of the remaining rock types tested were represented by sandstones of different porosities and hence different strengths of their cements. These were respectively Penrith sandstone (29% porosity), Darley Dale sandstone (13.5% porosity) and Tennessee sandstone (7.5% porosity). Each of these rock types has figured in previous rock mechanics investigations [Rutter and Mainprice, 1978; Cuss et al., 2003a, 2003b; Wong et al., 1997]. These sandstones lacked visible evidence of mechanical or textural anisotropy, and the wheels were fabricated in the plane of the bedding. Finally, an isotropically textured pink granite (of unknown provenance) was used.
 The unconfined compressive strength was measured for all six rock types using water-saturated cylindrical samples. Young's modulus and Poisson's ratio were also measured using cylindrical samples fitted with strain gauges in both axial and circumferential directions. For the special case of the muscovite-quartz schist and chlorite schist, prismatic strain gauged samples were used so that anisotropy of Poisson's ratio could be measured. The mineralogy of these rocks (from X-ray diffraction and optical studies) and physical properties are summarized in Table 1.
Table 1. Properties of the Rocks Tested
Darley Dale Sandstone
All rocks tested water saturated. Mechanical tests loaded parallel to foliation where appropriate. qtz, quartz; fsp, feldspar; biot, biotite; musc, muscovite; detr mica, detrital mica; pyx, pyroxene; hem, hematite.
1.0 ± 0.1
3.1 ± 0.3
2.4 ± 0.12
7.5 ± 0.3
13.5 ± 0.5
28.0 ± 0.5
Mineralogy solid phases
3% detr mica
Mean grain size, μm
1000 ± 200
450 ± 90
500 ± 90
75 ± 20
170 ± 50
130 ± 30
Unconfined compressive strength, MPa
168 ± 17
97 ± 10
188 ± 20
70 ± 7
38 ± 4
9.4 ± 1
Young's modulus, GPa
61 ± 6
38 ± 4
76 ± 8
8.9 ± 9
14.2 ± 2
7.2 ± 7
0.24 ± 0.05
0.31 ± 0.05
0.24 ± 0.05
0.27 ± 0.05
0.34 ± 0.06
0.29 ± 0.05
4.1. Calculation of Contact Stresses and Distance Traveled
 As the two wheels are pushed into together, their contacts are deformed elastically and a contact area forms according to the elastic properties of the rocks. The elastically softer rocks develop a larger contact area. According to the analysis of Hills et al. , if the width of the contact between two wheels of the same material is 2a, then a2 is given by
where P is the applied load (kN), r is the radius of the two wheels, ν is the Poisson's ratio, and E the Young's modulus of the rocks. The normal contact stress is in fact not homogeneous across the contact area, but the “mean” normal stress that has been taken to be representative of the contact is P/2ah, where h is the wheel thickness. Owing to the different elastic moduli for the various rock types, the contact stresses are different for each rock type for each of the 1 kg, 2 kg and 3 kg applied loads.
 The total distance d traveled by points on the wheel peripheries is
where t is total elapsed time, n is the combined number of wheel revolutions per second, T(t) is the amount of wear (combined reduction in wheel diameter) at time t, and ro is the starting diameter of the wheels. Although total wheel periphery displacements could be more than 10 km, a given small area 2ah is only under load once per revolution, so that the actual frictional contact at any given point is maintained only for a few tens of centimeters to a few meters of total displacement. The true contact displacement x is therefore given by
Although changes in the diameter of the wheels were taken into account in the calculation of shear displacement, the change in wheel diameter with time also implies that there will be a change in the value of the applied stress. For the changes in wheel diameters observed, this effect was calculated to be negligible. We have also neglected the effect of frictional traction at the wheel contacts on the value of the normal stress, because the modification of the average stress is relatively small (<5%) [Hills et al., 1993].
4.2. Wear Data
Table 2 shows experimental results for steady state wear for each applied load. Figure 4 shows plots of T versus x for each rock type studied under normal stresses corresponding to each of the applied loads. With exception of the data for the muscovite-quartz schist, a steady state wear rate was quickly established after the start of each run, and thereafter the wear rate was fairly linear. This implies that a steady state microstructure has been established at the periphery of each wheel. Only the muscovite-quartz schist showed significant evidence of an initial running-in phase, presumably corresponding to the finite displacement necessary to wear down initial surface irregularities and establish a steady microstructure. In each case, an increase in the value of applied normal stress produced an acceleration of the rate of wear. In all cases except Penrith sandstone the wear rate is very sensitive to small changes in the applied stress. Within experimental uncertainty, each wheel of a pair wore by the same amount.
Table 2. Experimental Results
Normal Stress, MPa
Wear Rate, mm m−1
Darley Dale Sandstone
 The low-porosity rocks (granite and the two metamorphic rocks) require the highest stress for a given wear rate and, given the scatter in the experimental data, behave in a similar fashion. The porous sedimentary rocks support lower stresses for the same applied loads owing to their generally lower elastic moduli, and also display higher absolute wear rates.
4.3. Particle Size Distributions of Wear Products
 Particle size distributions for wear products measured by laser particle sizer are presented in Figure 5. In each case the mean grain size of the intact rock is also shown. For all samples except the two porous sandstones, Penrith sandstone and Darley dale sandstone, the mean particle size is substantially lower than the intact rock mean grain size. Thus individual grains are broken down to rock flour by the abrasion process. In contrast, for the two porous sandstones, much of the wear debris is produced at the same mean grain size as the starting material. This suggests that the high porosity and relative weakness of the cement allows whole grains to be plucked out of the surface, with a minimal degree of granulation of the grains.
4.4. Microstructural Examination of Worn Wheels
 The worn wheels were examined visually (Figure 6). The low-porosity rocks (granite and the metamorphic rocks) displayed some degree of optical reflectivity, implying polishing to a surface finish of a few microns at most. This is consistent with the significant proportion of wear product particle sizes in the 1 to 10 μm range. There are also visible circumferential striations developed on the polished surfaces, produced through plowing by individual larger particles. The high-porosity Penrith and Darley Dale sandstone's surfaces remained rough and there was a high rate of wear, with wheel diameters being reduced by up to 16 mm. The wear involved whole grains being pulled out of the wheel surfaces, and this is confirmed by the particle size distribution results, which show peak concentrations of particles that are the same as the initial grain size.
 The edges of the wheels were also examined by scanning electron microscopy (SEM). Figure 7 shows the worn surface of a Tennessee sandstone sample. This rock demonstrates transitional behavior between polishing and plucking of whole grains. Grains that are well embedded in the rock matrix have been polished, to produce plateaus that are optically reflective yet display wear striations. On the other hand, a significant proportion of the surface displays irregular hollows lined with clay minerals, corresponding partly to original pore spaces and partly to cavities from which grains have been plucked.
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.
 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.
 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).
 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.
 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.
 We performed unconfined rock-on-rock abrasion experiments using a range of silicic rock types under different normal loads. Wear products were washed away by water, thereby simulating abrasion of a hard bed of a temperate glacier by entrained lithic clasts. An empirical wear law of the form
was obtained, in which A = 10−8.33, n = 8.33, and m = 4.50, with normal contact stress σ measured in MPa and ϕ is porosity. Wear rate dT/dx is expressed as m of wear per m of shear displacement of the clast. This relationship can in principle be used in an expression aimed at predicting rates of subglacial abrasion. In order to do this, it will be necessary to infer reliably clast/bed normal stresses and how horizontal clast velocity relates to basal ice sliding velocity. Provided significant (at least tenfold) enhancement of clast/bed normal stress can occur relative to the effective normal stress due to the ice overburden, our experimentally observed wear rates can correspond to naturally observed abrasion rates for temperate glaciers. We attribute the high value of the stress exponent in equation (11) to control of the wear rate by a subcritical crack growth process. Although we measured wear at only a constant displacement rate, it is possible that slower sliding rates might produce a given wear rate at a lower value of effective normal stress.
 In equation (11), porosity is used as a convenient parameter to describe strength contrasts between different silicic rock types. Low-porosity rocks wear through the production of silt-sized and finer-grained particles that are smaller than the initial grain size of the rock, and wearing surfaces become polished and striated. With increasing porosity, plucking of whole grains through intergranular cement failure becomes progressively more important, wear particle sizes are similar to the host rock grain size, and polishing becomes less well developed.
 This work was carried out while one of us (A.G.G.L.) was the holder of a UK NERC research studentship. Experimental Officer Robert Holloway constructed the apparatus used for the rock abrasion experiments. Nick Rutter (NOAA weather service) and David Collins (Salford University) allowed us to make use of their unpublished data on suspended sediment flux and borehole piezometric level for the Findeln Glacier. We are grateful to reviewers Chris Scholz and Chris Marone and to Associate Editor David Mainprice for constructive suggestions for improving the paper.