Introduction

The mechanical properties of pure-ice single crystals have been determined by a number of workers (i.e. Reference Griggs and GolesGriggs and Coles, 1954; Reference SteinemannSteinemann, 1954; Reference KambKamb, 1961; Reference Higashi, Koinuma and MaeHigashi and others, 1964, Reference Higashi, Koinuma and Mae1965; Reference Readey and KingeryReadey and Kingery, 1964; Reference Jones and GlenJones and Glen, 1968, Reference Jones and Glen1969; Reference MugurumaMuguruma, 1969; Reference Nakamura, Jones, Whalley, Jones and GoldNakamura and Jones, 1973; Reference Jones and BrunetJones and Brunet, 1978). Although the wide variety of testing conditions used makes it difficult to compare their results, several factors, including temperature, strain rate, initial dislocation density, previous straining history, surface condition and orientation have been shown to affect the mechanical behavior of ice single crystals. This paper is concerned with the effect of orientation on the strength of ice single crystals.

If yielding in a crystal depends only on when the applied stress, σ _a, resolved in the slip direction on the slip plane reaches a critical value, i.e. the critical resolved shear stress (CRSS), τ ^s, then Schmid’s law applies, i.e.

where θ is the angle between the loading direction and the normal to the slip plane, and λ is the angle between the loading direction and the slip direction. The term cos θ cos λ is often given the symbol, m, and referred to as the Schmid factor. τ ^s is a material property, which depends on the defect structure and the previous strain history. When yielding is controlled by a Peierls stress, the magnitude of τ ^s may depend on other components of the applied stress, such as the stress normal to the slip plane, which affect the dislocation core structure. In such cases, the use of Schmid’s law is inappropriate.

The structure of ice I_h consists of (0001) layers of water molecules stacked in the sequence AABBAABBAA. In principle, dislocations can glide either between widely spaced AA or BB planes (shuffle set) or between closely spaced AB or BA planes (glide set). Although it is still uncertain on which type of basal planes dislocations move, the perfect Burgers vectors, b, in both cases are .

Reference KambKamb (1961) concluded that, in ice single crystals, slip occurs in the direction of the maximum resolved shear stress acting across (0001), irrespective of the orientation relationship between b and σ _a. Thus, the resolved shear stress on the basal plane was determined by Reference Nakamura, Jones, Whalley, Jones and GoldNakamura and Jones (1973) and Reference Jones and BrunetJones and Brunet (1978) from

Most mechanical tests on ice crystals have been performed with the c axis around 45° to the loading direction. In other words, the effects of different orientations were not readily observable. However, Reference Brown and KawamuraBrown and Kawamura (1991) compressed sea-ice single crystals with c-axis orientations of 45° and 66° to the loading direction at −11 °C and a constant strain rate of 4 × 10⁻⁵ s⁻¹. They found peak stresses of 0.65 and 1.15 MPa, respectively, with corresponding steady-state flow (plateau) stresses of 0.31 and 0.45 MPa. As expected, the less favorable orientation for basal slip needed a larger stress for yielding and subsequent flow at the same axial strain rate. The value of τ ^s, calculated from Equation (1), at 66° was 0.37 MPa which was 32% higher than that (0.28 MPa) at 45°, while the subsequent resolved flow stresses were always slightly higher for the “66°-oriented” crystals. The difference in τ ^s may have been because stresses other than the resolved shear stress affect basal slip, or because the shear strain rates on the basal slip plane were different in the two tests.

In this paper we set out to answer the question, how does the crystallographic orientation affect the flow of ice single crystals? In other words, is Schmid’s law applicable to ice?

Experimental

A Nalgene tank with a 19 L capacity, filled with distilled water, was placed in a cold room at1°C for 1 day to equilibrate the water temperature before it was moved into a −4°C freezer for ice growth. The sides and bottom of the tank were wrapped with Armaflex for insulation so that the ice grew only from the top downwards. Generally, it took 7 days to grow an ice puck about 13 cm thick. During the ice growth the tank was left uncovered. Using this method, columnar ice with a large grain-size was obtained. The long axis of the columns was along the growth direction, while the c axis was randomly oriented in the plane perpendicular to the growth direction.

After the ice puck was removed from the tank, the top and bottom were removed using a chainsaw. A band-saw was then employed to make the remaining ice rectangular, and the rough surfaces were shaved with a razor blade. The ice was stored in a plastic bag in a −10°C freezer for later use.

Reference Higashi, Koinuma and MaeHigashi and others (1965) first observed that the steady-state strain rate in bending creep tests was affected by the surface condition of ice single crystals. This effect was further studied in constant-strain-rate compression tests by Reference MugurumaMuguruma (1969). For example, at −10°C and a constant strain rate of 4.2 × 10⁻⁵ s⁻¹, the peak stress for chemically polished specimens was approximately double that for mechanically polished specimens, while the latter was more than twice that for crystals containing sub-boundaries. The variation in the shape of stress–strain curves was considered to be due to differences in the dislocation density of crystals. The imperfect surface produced by mechanical polishing and the sub-boundaries acted as sources of dislocations. In order to avoid the effects of surface conditions on the mechanical behavior of ice single crystals, all specimens used here were prepared in exactly the same way. Further sample preparation utilized the following steps:

1. (1) Polarized illumination was used to examine the grain-size in the ice bulk. Under the polarized light, each grain showed different contrast due to different orientations (Reference HobbsHobbs, 1974). Several large grains were marked and then cut out using the band-saw. After cutting, the rough surfaces were shaved with a razor blade.

2. (2) The orientation of each ice single crystal was determined using the etch-pit technique to ±2° (Reference HiguchiHiguchi, 1958). After the c axis and a axes were identified according to the shapes of etch pits, polarized light was employed to further confirm the c axis to ±1°.

3. (3) Knowing the orientation of the ice single crystal, the specimen was cut so that the basal plane (0001), marked by a hexagon in Figure 1, was tilted an angle θ from the top surface, with planes on the side. The and directions are also indicated in Figure 1. The values of θ were chosen to be in the range 3–70°, with several specimens tested at some orientations.

4. (4) A horizontal-milling machine housed in a cold room was used to smooth the surfaces of specimens to their final dimensions of about 20 mm×25 mm×60 mm.

5. (5) The dimensions of all the specimens were measured using a caliper ruler, and their orientations were again determined under polarized light to ±1°.

Fig. 1. Schematic showing the dimensions and orientation of the pure-ice single crystals. The plane with the hexagon is the basal plane. The basal plane was tilted from θ the top surface. and axes are also indicated.

A series of constant-strain-rate compressive tests was performed on these ice single crystals as a function of orientation at −20±0.5°C at a constant axial strain rate of 1 × 10⁻⁵ s⁻¹ using a servohydraulic Material Testing Systems device housed in a cold room. A limited number of compression tests were also performed at −10±0.5°C for crystals with orientations where θ = 5°±1° at strain rates from 1 × 10⁻⁶ s⁻¹ to 1 × 10⁻⁴ s⁻¹.

Results and Discussion

Typical engineering stress–strain curves at −20°C for various orientations obtained at a constant axial strain rate of 1 × 10⁻⁵ s⁻¹ are shown in Figure 2. Except for crystals with θ = 3° and 5°, their shape is generally similar, i.e. the stress rises linearly to a peak stress after which it falls rapidly and then more slowly towards a plateau. The specimen with θ = 3° fractured immediately after the peak stress, and the test at θ = 5° was stopped at an axial strain of about 2% (when the stress was close to zero). All other tests were continued up to an axial strain of about 5%.

Fig. 2. Typical engineering stress-strain curves of pure-ice single crystals at an axial strain rate of 1.0 × 10⁻⁵ s⁻¹ and −20° C, with orientations: (a) θ = 3–10°; (b) θ = 15–33°; and (c) θ = 50–70°.

Figure 3 shows the peak stress (yield stress), obtained from θ = 3–70°, plotted as a function of the Schmid factor, cos λ cosθ. The peak stress depends on the initial mobile dislocation density. Here it is assumed that all the specimens had similar dislocation densities since, as noted earlier, they were made in exactly the same way. The reproducibility of the results at a particular orientation tended to confirm this (see Fig. 3).

Fig. 3. Peak stress (at θ = 3−70°) vs Schmid factor (cos λ cos θ) at a constant axial strain rate of1.0 × 10⁻⁵ s⁻¹. When more than one test was performed at a particular orientation and the data points overlap, the number of tests is indicated.

Although the peak stress values in Figure 3 were obtained at the same axial strain rate, the shear strain rate on the basal plane, , varied considerably as θ varied from 3° to 70°. In order to determine whether Schmid’s law is obeyed in ice it is necessary to compare the GRSS values at different values of cos θ cos λ at the same shear strain rate. Thus, was calculated by dividing the shear strain on the basal slip plane, γ, by the time. γ is given by (Reference Schmid and BoasSchmid and Boas, 1950; Reference ReidReid, 1973)

where ℓ₀ is the original length of the specimen, L is the deformed length of the specimen, λ₀ is the initial value of λ (at the start of the test) and θ ₀ is the initial value of θ. Figure 4 is a plot of the calculated GRSS normalized to a constant shear strain rate of 1 × 10⁻⁵s⁻¹ at –20°C vs the Schmid factor. To normalize the CRSS, the experimentally determined relationship between the peak stress, τ, and the shear strain rate was used (Reference Trickett, Baker and PradhanTrickett and others, in press). If Schmid’s law is obeyed, the CRSS at a constant shear strain rate will be independent of Schmid factor. Within experimental error, Figure 4 shows that this is indeed the case for ice.

Fig. 4. Calculated peak CRSS (at θ = 3−70°) normalized to a constant shear strain rate on the basal slip plane of 1.0 × 10⁻⁵ s⁻¹ at −20^o C vs the Schmid factor (cos λ cos θ).

That yielding of single crystals depends on stresses other than the CRSS obtained from Schmid’s law has been noted in a number of materials. For example, the mechanical behavior of NiAl strongly depends on its crystallographic orientation (Reference Takasugi, Kishino and HanadaTakasugi and others, 1993; Reference Mielec, Novak, Zarubova and GemperleMielec and others, 1997), with the calculated CRSS of stoichiometric NiAl increasing with increasing compressive stress. It was suggested that this result is due to the effect of the normal stress on the dislocation core structures. For close-packed materials, i.e. face-centered cubic (f.c.c.) and hexagonal close-packed (h.c.p), we do not expect any effect of stresses other than the resolved shear stress, since atoms behave like hard spheres, i.e. they are incompressible. In f.c.c. metals, the multiplicity of equivalent slip systems makes it impossible to test Schmid’s law over a wide range of orientations. In h.c.p. metals, in contrast, Schmid’s law has been shown to apply over a large range of orientations (Reference HoneycombeHoneycombe, 1968).

Typical engineering stress–strain curves for specimens with θ = 5° obtained at −10°C and with various axial strain rates are shown in Figure 5. In this hard orientation, the crystals tested at 1.0 × 10⁻⁴ s⁻¹ failed by brittle fracture. Using polarized light, the cleavage cracks in the loading direction were determined to have propagated on {1010} planes (see Fig. 6). The onset of cracking was observed at a strain of 0.7%, corresponding to 60% of the peak stress.

Fig. 5. Typical engineeringstress-strain curves at different axial strain rates of pure-ice single crystals with c axes at 5° to the loading direction at −10°C.

Fig. 6. Slip bands (the almost horizontal lines) in a θ = 5° specimen tested at –10°C and a constant axial strain rate of 1.0 × 10^{−6 − 1}. The vertical cracks grew on {1010} planes.

Evenly spaced, well-defined slip bands were observed in these strained specimens (see Fig. 6). Thus, it was found that even at a hard orientation, i.e. with the basal plane tilted about 5° from the top surface, the macroscopic plastic deformation was still produced by basal slip. Further, both the presence of the well-defined slip bands and the sharp peaks in the stress–strain curves suggest that the initial dislocation density was low (Reference MugurumaMuguruma, 1971; Reference Brown and KawamuraBrown, 1991). Note that slip lines could not be observed when testing at faster strain rates at −10° or −20°C.

Conclusions

Stress–strain curves were obtained from laboratory-grown ice single crystals under compression at −20°C and an axial strain rate of 1.0 × 10⁻⁵ s⁻¹. It was shown that the critical resolved shear stress (normalized to a constant shear strain rate of 1 × 10⁻⁵ s⁻¹ ) for basal slip obeys Schmid’s law over a wide range of orientations. Slip was shown, using slip lines, to occur on basal planes.