1. Introduction

In modeling failure of the arctic sea-ice cover, Reference Schreyer, Sulsky, Munday, Coon and KwokSchreyer and others (2006) invoked frictional sliding by incorporating a Coulombic failure envelope. In so doing, they questioned whether, in addition to traction on the plane of material failure (i.e. in addition to the normal (σ_nn ) and shear (σ_nt ) components of traction; Fig. 1), the normal component of stress, σ_tt (Fig. 1), that is oriented parallel to the sliding plane might affect the sliding resistance. The question is not limited to the arctic sea-ice cover, but applies as well to tectonic evolution of icy bodies of the outer solar system, such as Jupiter’s moon Europa (Reference Tufts, Greenberg, Hoppa and GeisslerTufts and others, 1999; Reference KattenhornKattenhorn, 2004) and Saturn’s satellite Enceladus (Reference Nimmo, Spencer, Pappalardo and MullenNimmo and others, 2007; Reference Smith-Konter and PappalardoSmith-Konter and Pappalardo, 2008). Reference Schreyer, Sulsky, Munday, Coon and KwokSchreyer and others (2006) did not expand upon the underlying physics, although one could imagine that under all-compressive loading, the stress component σ_tt might act to prevent either formation or propagation of surface cracks inclined to both the sliding plane and sliding direction, of the kind reported by Reference Montagnat and SchulsonMontagnat and Schulson (2003).

Fig. 1. Schematic diagram showing orientation of stresses with respect to sliding plane. σ ₁ and σ ₂ are the principal stresses with respect to the loading system, σ_nn is the normal stress perpendicular to the sliding plane, σ_tt is the normal stress parallel to the sliding plane in the direction of sliding and σ_nt is the shear stress on the sliding plane.

To determine whether this third component of stress (where the tensor is expressed in terms of a coordinate system defined by the failure plane and by the direction of sliding) actually affects the measured resistance to sliding, a series of experiments was performed in which the two normal components of the stress tensor were varied independently. The results are reported here. To our knowledge, this is the first report to examine this point experimentally.

2. Experimental Procedure

We chose to examine columnar-grained, polycrystalline ice that possesses the S2 (Reference Michel and RamseierMichel and Ramseier, 1971) growth texture, and to slide the ice across itself, column against column, in a direction normal to the long axis of the grains, in the manner that a sheet of first-year sea ice might slide across a closed lead or Coulombic fault (Reference SchulsonSchulson, 2004). For simplicity, we examined freshwater ice, since there is little to distinguish the Coulombic failure envelope of the two kinds of material (Reference Schulson, Fortt, Iliescu and RenshawSchulson and others, 2006a). We made the ice in the laboratory by unidirectionally freezing local tap-water, and then verified its microstructure, as described elsewhere (Reference Fortt and SchulsonFortt and Schulson, 2007). Subsequently, we prepared plate-shaped specimens of dimensions 160 mm × 80 mm × 50 mm, with the long axis of the columnar grains oriented perpendicular to the largest faces. To prepare a sliding interface, we cut the specimens diagonally from corner to corner, as sketched in Figure 2, and then polished the exposed faces by gently rubbing using a warm, optically flat glass plate on a lapping stone. The surface roughness in the direction of sliding was obtained using a stylus prof ilometer and found to be (4 ± 12) × 10⁻⁶ m. To hold together the two wedge-shaped blocks (Fig. 2) during pre-loading, holes of 5 mm diameter were drilled along the X ₃ direction and joined by a piece of string. To allow sliding without crushing the apex of each wedge, shims of chemically polished brass were placed on the top and the bottom of the wedges (Fig. 2). To reduce friction along the ice–shim interface, thin (0.15 mm) polyethylene sheets were inserted.

Fig. 2. Schematic diagram showing experimental set-up of (a) 26° oriented sliding plane test, and (b) 64° oriented sliding plane test.

We oriented specimens for sliding in one of two ways. In the first, the long axis of the specimen was oriented parallel to the X ₁ axis (Fig. 2a) and the sliding plane was inclined by θ ∼ 26° to X ₁, but parallel to the direction of the columns, X ₃. In the second, the long axis of the specimen was oriented parallel to X ₂ (Fig. 2b) and the sliding plane was inclined by θ ∼ 64° to X ₁. In both cases, the specimens were loaded under biaxial compression where the major stress, σ ₁, was applied along direction X ₁ and the minor stress, σ ₂, was applied along direction X ₂, using a true multiaxial loading system housed within a cold room of Dartmouth’s Ice Research Laboratory. Assuming frictionless contact between the ice and loading shims, the ratio, χ, of the normal stresses parallel to the plane of sliding σ_tt and perpendicular to the plane of sliding σ_nn is given by

(1)

For the geometry of the two set-ups and for the values measured for the two applied stresses (given below), this parameter varied by about a factor of two, from χ ∼ 1.4 for θ ∼ 26° to χ ∼ 0.65 for θ ∼ 64°.

To effect sliding, a constant displacement rate, V _A, was applied in the X ₁ direction and then converted to sliding speed, V _S, along the inclined plane through the expression

(2)

For the two different orientations of the sliding plane, the applied velocity was such that the sliding speed in both cases was V _S = 8 × 10⁻⁴ m s⁻¹. The velocity was chosen as it is within the range of velocities observed in the arctic ice pack (Reference Moritz, Stern, Dempsey and ShenMoritz and Stern, 2001). The ice was slid a total distance of 8 mm, recorded by extensometers that were attached to the loading platens. In all tests the temperature of the ice was −10°C. During each test, the loads applied in the two directions X ₁ and X ₂ were recorded electronically at a rate of 1000 scans s⁻¹ and then converted to the principal stresses σ ₁ and σ ₂. Loads were measured with a sensitivity of ±100 N, which translated to a sensitivity in principal stress of ±0.03 MPa on the smaller faces and ±0.015 MPa on the larger faces. The experiments were performed such that the minor stress, σ ₂, was held constant for the duration of each test and the range of normal stresses tested was chosen to reflect that observed in the arctic ice pack (albeit on the lower side of our range).

3. Results

We performed a total of 18 experiments, 11 for the orientation of the sliding plane θ ∼ 26°and 7 for θ ∼ 64°. Figure 3 shows examples of load-displacement curves from which the appropriate stresses were computed (using the midpoint of the load at each displacement). Table 1 lists the results. Listed in the table are the computed values of the two normal stresses and the computed value of the shear stress, σ_nt , after sliding different amounts (0, 2.4 mm, 4 mm, 8 mm).

Table 1. Experimental results. Test No. indicates the Ice Research Laboratory test label, θ is the measured angle of the sliding plane with respect to X ₁, δ _S is the sliding displacement, σ_nn is the normal stress perpendicular to the sliding plane, σ_nt is the shear stress on the sliding plane and σ_tt is the normal stress parallel to the sliding plane. Italics signify the 64° tests, whereas the 26° tests are in regular font.

Fig. 3. Examples of stress-vs-sliding-displacement curves for (a) 26° oriented sliding plane test and (b) 64° oriented sliding plane test.

Figure 4 plots the shear stress on the sliding plane versus the normal stress acting across the sliding plane, after displacement, δ _S, of 0, 2.4, 4 and 8 mm, for both orientations of the sliding plane. The displacements were chosen to enable comparisons to be made with our previous work (Reference Fortt and SchulsonFortt and Schulson, 2007). The curves may be described reasonably well by straight lines (coefficient of determination R² shown in the figure), in keeping with earlier measurements of frictional sliding of ice sliding slowly upon itself under a relatively low normal stress (Reference Fortt and SchulsonFortt and Schulson, 2007). The coefficient of friction is given by the slope of the curves (more below) and appears to be higher (μ = 0.27) and more variable at the onset of sliding than during sliding where it increases from μ = 0.16 at a displacement of δ _S = 2.4 mm to μ = 0.21 at δ _S = 8 mm. The implication is that the static coefficient of friction is greater than the kinetic coefficient. This is not surprising given the added time for sintering before the test starts. There also appears to be a systematic trend of increasing friction with increasing displacement. However, we prefer not to place too fine a point on this trend until further work is done. of particular interest from the perspective of these experiments is the observation that within experimental uncertainty the data for the two orientations fall upon the same line for each displacement. This implies that under the present conditions the normal component of the stress tensor parallel to the direction of sliding has no systematic effect on the resistance to sliding, at least within the sensitivity of the measurements.

Fig. 4. Graph of σ_nt vs σ_nn at four sliding displacements. Black (solid) points indicate 26° data points; white (open) points indicate 64° data points.

4. Discussion

The observation that the shear stress at ‘failure’ scales linearly with the normal stress across the sliding plane is not surprising. Such behavior has been observed earlier for ice sliding slowly across natural Coulombic shear faults (Reference Schulson, Fortt, Iliescu and RenshawSchulson and others, 2006b; Reference Fortt and SchulsonFortt and Schulson, 2007) at the speed and temperature applied in the present tests. The sliding resistance thus obeys Coulomb’s failure criterion

(3)

where, again, σ _nt is the shear stress on the sliding plane, μ the kinetic coefficient of friction and σ _nn the normal stress across the sliding plane; σ₀ is the cohesive strength. Barring the variations with displacement noted above and averaging over the results over all displacement, σ₀ = 0.02 ± 0.02 MPa and μ = 0.20 ± 0.03, in agreement with values derived by Reference Kennedy, Schulson and JonesKennedy and others (2000) and by Reference Montagnat and SchulsonMontagnat and Schulson (2003) from double-shear experiments on the same kind of ice under similar experimental conditions.

A modification of Coulomb’s criterion, which includes the other normal stress, may be written as

(4)

where κ denotes the sensitivity of the sliding resistance to the normal stress parallel to the sliding direction. The observation that there is essentially no effect of this stress component, namely that dσ _nt /dσ _tt ∼ 0, suggests that this component is not a significant factor in sliding resistance under the conditions tested here. Therefore, under the present conditions κ ≈ 0.

We do not know whether κ ≈ 0 under other conditions. Friction of ice is a complicated property and depends upon both sliding speed and temperature. For instance, at higher sliding velocities (V _S >10⁻⁵ m s⁻¹ at −10°C), ice exhibits velocity weakening, evident from the fact that the kinetic coefficient of friction decreases with increasing sliding speed (Reference Kennedy, Schulson and JonesKennedy and others, 2000; Reference Maeno, Arakawa, Yasutome, Mizukami and KanazawaMaeno and others, 2003; Reference Fortt and SchulsonFortt and Schulson, 2007). Under such conditions, localized melting appears to play an important role (Reference Kennedy, Schulson and JonesKennedy and others, 2000; Reference Hatton, Sammonds and FelthamHatton and others, 2009). Our experiments were performed within this velocity-weakening regime, and so melting could perhaps account for the apparent absence of an effect of σ_tt . At lower sliding velocities (V _S < 10⁻⁵ m s⁻¹), the character of sliding changes: sliding resistance exhibits velocity strengthening (Reference Fortt and SchulsonFortt and Schulson, 2007), and the governing mechanism appears to be localized creep (Reference Kennedy, Schulson and JonesKennedy and others, 2000; Reference Fortt and SchulsonFortt and Schulson, 2007). Whether an effect of σ_tt might be detectable within the velocity-strengthening regime remains to be seen. There is also a question of whether the sliding behavior of sea ice or of extraterrestrial ice differs from that of freshwater ice, owing to the presence within sea ice (Reference WeeksWeeks, 2010) of additional phases (brine, air and, at lower temperatures, precipitated salts) and (possibly) within extraterrestrial ice of hydrated salts (Reference McCordMcCord and others, 1998). Although we cannot offer a firm answer, our sense is that the similarity here may be greater than the difference, at least in the case of sea ice where the friction coefficient of the two materials within the velocity-weakening regime is almost identical (Reference Kennedy, Schulson and JonesKennedy and others, 2000). Finally, one might wonder whether spatial scale is a factor. Again, we cannot offer a firm statement. However, given that the brittle compressive failure envelope of the arctic sea-ice cover has the same slope as one generated in the laboratory from specimens harvested from the winter sea-ice cover (Reference Weiss, Schulson and SternWeiss and others, 2007), and given that the slope of the brittle compressive failure envelope is governed by the friction coefficient (Reference Schulson, Fortt, Iliescu and RenshawSchulson and others, 2006b), our sense is that the character of sliding within the sea-ice cover is probably similar to that within test specimens. Clearly, more work is needed before the generality of the present finding can be assessed.

5. Conclusion

At this juncture, therefore, we conclude that there is no evidence that the resistance to sliding of warm (−10°C), freshwater ice upon itself at a relatively low speed (8 × 10⁻⁴ m s⁻¹) under low normal stresses (< 1.2 MPa) is affected by the component of normal stress parallel to the direction of sliding.