Uniaxial compression experiments

A series of 12 unconfined uniaxial compression experiments was conducted on cylindrical samples of the laboratory- made initially isotropic polycrystalline ice. The samples were machined to a nominal diameter, D = 25.4mm, using a lathe, had initial heights z₀ = 70−80 mm and a mean grain area of ˜4mm². The experiments were conducted at three constant octahedral shear stresses, t_o = 0.20, 0.40 and 0.80MPa at −2.0°C. Stresses and strain rates are expressed in terms of the octahedral shear stress, t_o, and strain rate, , which are defined in the Appendix.

Transverse extension of cylindrical samples during vertical compression leads to barrelling at large strains (Poisson effect), which increases the sample cross-sectional area and reduces t_o. To minimize variations in t_o the load was periodically increased during experiments at t_o = 0.20 and 0.40MPa, based on the accumulated vertical strain. The additional load necessary to maintain t_o was estimated by assuming incompressibility and that transverse strains are uniform along the sample length, leading to a uniform increase in sample cross-sectional area. Despite the use of this technique, the potential for errors in t_o increases as strains exceed 10%. To further reduce uncertainty in t_o the samples were removed from the apparatus and machined back to a nominal diameter of D = 25.4 mm, which allowed the load to be accurately reset prior to recommencing the experiment. A similar two-stage experimental procedure was used by Reference Gao and JackaGao and Jacka (1987) to provide more accurate control of t_o when total strains exceed ˜20%. This procedure takes less than an hour, was conducted at −18°C to minimize microstructural modification due to stress relaxation and provides the opportunity to cut a horizontal thin section from the middle of the sample for mid-experiment crystallographic examination. Following removal of a thin section, precision machining of the end-faces with a lathe enabled the two sections of the original sample to be easily rejoined.

Tertiary creep strain rates at strains ε_o > 10% were determined for the 12 experiments commenced on initially isotropic samples. Following the sectioning and machining procedure (only for experiments at t_o = 0.20 and 0.40 MPa) a further eight tertiary creep rates were determined at strains exceeding 10%. These particular experiments were commenced on samples with initially anisotropic small- circle girdle crystal orientation fabrics, which are compatible with the uniaxial compression stress configuration.

Simple-shear experiments

Horizontal shear experiments were conducted on samples with initially anisotropic crystal orientation fabrics obtained from the 1196m Dome Summit South (DSS) ice core, drilled 4.7 km from the summit of Law Dome, East Antarctica (Reference Morgan, Wookey, Li, Van Ommen, Skinner and FitzpatrickMorgan and others, 1997). Additional initially isotropic samples were prepared from laboratory-made material. All shear experiment samples were prepared using a bandsaw, and dimensions varied according to the amount of material available from the selected ice-core sections. Typically, samples were 85-100mm long and 15−20mm wide and high.

Reference Russell-HeadRussell-Head and Budd (1979) and Reference Gao and JackaGao and Jacka (1987) have shown experimentally that where deformation samples have an initial crystal orientation fabric compatible with the applied stress configuration, the strain rate decreases directly to the steady-state value, ct_er, i.e. the minimum strain rate for samples with the compatible fabric is the steady-state tertiary rate. This creep behaviour is illustrated schematically by the upper curve in Figure 1. We used a similar experimental technique to obtain pseudo-tertiary-creep rates at strains of 1-3% for DSS ice-core samples with pre-existing anisotropic crystal orientation fabrics.

Individual samples from three distinct depth zones within the DSS core (DSS 107 (146 m), DSS 521 (550 m) and DSS 1012 (963 m) - depths are from the surface to the top of the core sections) were used to investigate the influence of crystal orientation fabric strength on the magnitude of strain-rate enhancement at t_o = 0.1 MPa. Crystal orientation fabrics measured prior to deformation are presented in Figure 3. Repeat measurements of the DSS borehole inclination (Reference Morgan, Van Ommen, Elcheikh and LiMorgan and others, 1998) and flow modelling of the drill site (Reference Wang, Warner and BuddWang and others, 2002) confirm the DSS 1012 sample is from a high-shear region, ˜230m above bedrock. This sample was selected to provide an estimate of the maximum strain-rate enhancement resulting from simple shear on the compatible strong single-maximum crystal orientation fabric. Based on surface deformation measurements, compression at the DSS borehole site is nearly uniaxial: the longitudinal (in the line of flow) strain rate of 3.2 × 10˜⁴a˜¹ is slightly lower than the transverse strain rate of 4.5 × 10˜⁴ a˜¹ (Reference Morgan, Van Ommen, Elcheikh and LiMorgan and others, 1998). Modelling by Reference Wang, Warner and BuddWang and others (2002) suggests a vertical compression-dominated stress configuration at 146 m for DSS 107 and a combined shear and compression configuration at 550 m for DSS 521. Thus, unlike the DSS 1012 sample, the experimental simple-shear stress configuration does not match the in situ stress configuration for the DSS 107 and DSS 521 samples. Strain rates for these samples will indicate the relative magnitude of crystal orientation fabric strength on strain-rate enhancement.

Experiments by Reference Gao and JackaGao and Jacka (1987) and Reference Li and JackaLi and Jacka (1998) on samples with initially anisotropic crystal orientation fabrics not compatible with the applied stress configuration show that the minimum strain rate at ˜1% strain differs from the value for initially isotropic samples. For the majority of high-strain deformation in polar ice masses a dynamic balance exists between deviatoric stress, strain rate and fabric evolution. Therefore, while incompatible combinations of anisotropy and stress configuration do not replicate in situ conditions within natural ice masses, deformation experiments involving such combinations allow the influence of fabric on strain-rate enhancement to be assessed.

Uniaxial-compression (Reference JackaJacka and Maccagnan, 1984) and simple-shear (Reference GaoGao and others, 1989) experiments on initially isotropic polycrystalline ice samples have shown that at strains below ε_o ≈2−3% microstructural and strain-rate changes relative to the minimum isotropic state are minimal. Based on these results, the total accumulated strain for experiments where the samples had a pre-existing anisotropy was restricted to 1−3%, in order to limit microstructural evolution and ensure the measured strain rates correspond to the initial fabric.

A series of seven simple-shear experiments at t_o = 0.075, 0.10 and 0.20MPa on isotropic laboratory-made ice were used to determine control values for the isotropic minimum strain rate. For these experiments and those on DSS ice-core samples, a temperature-cycling procedure was used to obtain strain-rate estimates at various temperatures. Experiments were commenced at −5°C. Once the minimum strain rate had been reached, at ε_o ≈ 1%, the temperature was cycled down through −10 and −15°C before cycling in reverse through the same temperatures and −2 ^° C. Once the samples had equilibrated to each new temperature set point, it was found that a minimum accumulated strain of 0.1% was adequate to achieve a stable strain rate prior to the next temperature step. Reference Russell-HeadRussell-Head (1979) has shown that for a given temperature, minimum strain rates determined while stepping down through a range of temperatures are equivalent to those when stepping up through a temperature range, and are also equivalent to those determined in experiments conducted at a single constant temperature. A further two simple-shear experiments were conducted on initially isotropic laboratory-made ice at t ₀ = 0.4MPa and −2.0°C where the total accumulated strain exceeded 10%, allowing development of stable tertiary creep strain rates.

At the conclusion of all experiments, horizontal thin sections were cut from the samples for mean grain area and crystal orientation fabric analysis. Crystal orientation fabrics were measured using a Russell-Head Instruments G- 50 automated ice crystal fabric analyser. Fabric data are presented for a horizontal reference frame using lower- hemisphere projection Schmidt equal-area plots. The mean grain area was determined from the total number of grains within a thin section and the thin-section area. The total number of grains was determined during crystal orientation fabric measurement, and the pixel count for the corresponding thin-section bitmap image (produced by the automated fabric analyser) was determined via digital image analysis. The bitmap pixel dimensions are 43 µm×43µm.

Uniaxial compression experiments

Representative creep curves for uniaxial compression experiments at t ₀ = 0.20, 0.40 and 0.80 MPa are presented in Figure 4. Distinct minima in for experiments commenced on initially isotropic ice are visible at ˜1% strain. In the two-stage experiments at t ₀ = 0.20 and 0.40MPa, c_o decreases directly to the tertiary creep rate during the second stage, as samples have pre-existing small-circle girdle crystal orientation fabrics. Crystal orientation fabrics measured at the conclusion of uniaxial compression experiments at t ₀ = 0.20 and 0.40MPa, where the total accumulated strain was ε_o > 20%, are presented in Figure 5.

Fig. 4. Octahedral shear strain rate as a function of octahedral shearstrain for uniaxial compression experiments at τ _o = 0.20, 0.40and 0.80MPa and −2°C. Representative data from experiments on initially isotropic samples and those with an initial small-circle girdle crystal orientation fabric are presented. A distinct minimum in strain rates occurs for initially isotropic samples. For experiments commenced at τ _o = 0.20 and 0.40MPa, on samples with an initially anisotropic small-circle girdle crystal orientation fabric, strain rates decrease directly towards the steady-state tertiary value.

Fig. 5. Crystal orientation fabric data measured after the completion of uniaxial compression experiments at −2ºC. (a) τ _o = 0.20MPa, (b) τ _o = 0.40MPa. (Crystal orientation fabric data as Fig. 3.)

The effect on strain rates of increasing the compression load to account for sample barrelling is most apparent in experiments at t ₀ = 0.40MPa (Fig. 4). Distinct step increases in strain rates at strains ≳3% indicate where the load adjustments were made. A gradual decline in strain rates, due to a continued decrease in t ₀ with accumulating strain, is evident between load adjustments. Step changes in strain rates are smaller for experiments at t ₀ = 0.20MPa, as the required load increments are smaller than those for t ₀ = 0.40 MPa and the lower strain rates allow more frequent load adjustments as a function of strain.

Isotropic minimum and steady-state tertiary creep strain rates for all compression experiments are shown in Figure 6. From linear regression of the data, 95% confidence intervals were determined for the creep power- law stress exponent (n in Eqn (2)); n = 3.8 ± 0.3 for steady- state tertiary strain rates and n = 3.3 ± 0.3 for isotropic minimum strain rates. The higher value of n for tertiary creep compared with that for (secondary) minimum creep can be interpreted as a stress-dependent level of strain-rate enhancement, where , such that enhancement increases from E ≈ 2.8 at τ_o = 0.20 MPa to E ≈ 6.3 at τ₀ = 0.80 MPa.

Fig. 6. Isotropic minimum and steady-state tertiary creep octahedral shear strain rates for uniaxial compression experiments at τ _o = 0.20, 0.40 and 0.80MPa and −2°C. Strain-rate enhancement, increases as a function of τo. Limits of the creep powerlaw stress exponent, n, are 95% confidence intervals. The dottedlines illustrate the small difference in the isotropic minimum and tertiary creep rates when adjusted to stress exponents of n = 3.0 and n = 3.5, respectively. The widely accepted stress exponent of n = 3 for the minimum creep rate of isotropic polycrystalline ice isdiscussed in the text, while n = 3.5 for steady-state tertiary creep is based on simple-shear experiments (Fig. 9) conducted over a wider range of stresses.

Simple-shear experiments

Minimum strain rates for the simple-shear experiments on initially isotropic ice at temperatures from —2 to −20°C (Fig. 7) illustrate the significant effects of stress and temperature on isotropic polycrystalline ice flow. Minimum isotropic strain rates from the similar simple-shear experiments of Reference Russell-HeadRussell-Head and Budd (1979) have been included to provide additional data at each temperature, and to extend coverage down to shear stresses t_o = 0.022 MPa. Creep power-law stress exponents of n≈ 3.0 determined from the minimum isotropic strain-rate data are consistent with other compilations of minimum isotropic strain rates (e.g. Reference Budd and JackaBudd and Jacka, 1989).

Fig. 7. Minimum octahedral shear strain rates for simple-shear experiments on initially isotropic laboratory-made polycrystalline ice. Additional data are from Reference Russell-HeadRussell-Head and Budd (1979). Solid lines indicate a least-squares fit to data for each temperature. Limits of n, the creep power-law stress exponent, for each dataset are 95% confidence intervals. Alternating black and grey symbols and lines distinguish adjacent datasets at different temperatures.

The minimum strain rates determined for DSS samples with initially anisotropic crystal orientation fabrics at strains of ε _o = 1-3% vary according to the nature of the pre-existing fabric (Fig. 8). These strain rates can be compared via enhancement factors calculated using isotropic minimum strain rates from the control experiments (Fig. 7) as a reference. In general, enhancement decreased for increasing φ _1/4 values, where φ _1/4 is the cone angle containing the first quartile of c-axis colatitudes. The highest mean enhancement, E = 5.2, occurs for the DSS 1012 (963 m) ice-core sample obtained from the high-shear layer identified in the DSS borehole (Reference Morgan, Van Ommen, Elcheikh and LiMorgan and others, 1998). As simple shear is the in situ stress configuration for this sample, E = 5.2 provides an estimate of the maximum strain-rate enhancement for shear on a strong single-maximum crystal orientation fabric at τ _o = 0.10MPa and temperatures from −2 to −15ºC. The crystal orientation fabrics for the DSS 521 and DSS 107 samples are only partially compatible with the experimental simple-shear stress configuration. The lower mean E values for these samples are related to their reduced proportion of near-vertical c-axes in comparison with the DSS 1012 sample (Fig. 3), which is indicated by generally higher φ _1/2 and φ _1/4 values. Unlike the effect of stress on enhancement, Figure 8 indicates temperature does not influence the level of strain-rate enhancement.

Fig. 8. Octahedral shear strain rates for horizontal simple-shear experiments on initially isotropic laboratory-made polycrystalline ice and initially anisotropic samples from the DSS ice coreconducted at τ _o = 0.10MPa. Additional isotropic minimum strainrate data are from Reference Russell-HeadRussell-Head and Budd (1979). Lower values of φ _1/4 indicate fabrics that are more strongly vertically clustered.

As the simple-shear tertiary creep data from the current study are limited, results from previous experiments conducted using the same or similar apparatus have been included to further illustrate differences in isotropic minimum and steady-state tertiary creep rates as a function of stress at −2 ^° C. The sources of additional data are indicated in Figure 9. Due to the scarcity of low-stress tertiary creep data, additional data from experiments by Reference LileLile (1984) at −6°C have been included after rescaling to −2°C based on strain-rate, stress and temperature relationships determined from the isotropic minimum creep data of Figure 7. For stresses between τ₀ = 0.04 and 0.40 MPa the tertiary creep power-law stress exponent in simple shear (n = 3.5 ± 0.5) is higher than that for isotropic minimum strain rates (n = 3.0 ± 0.2). These data suggest the enhancement−stress proportionality of determined from uniaxial compression experiments, also applies to simple shear; however, the magnitude of E is greater for simple shear. Analysis of the data in Figure 9 indicates enhancement increases from E = 2.9 at t_o = 0.04MPa to E = 9.1 at To = 0.40 MPa. 10

Fig. 9. Tertiary creep octahedral shear strain rate, (open symbols), vs octahedral shear stress, τ _o, for simple-shear experiments. The corresponding isotropic minimum strain rates (filled symbols) are presented for comparison. All experiments were conducted at −2°C using the same or similar apparatus. Sources of additional experimental data are listed in the legend. Limits for the creep power-law stress exponents, n, for isotropic minimum and steady- state tertiary creep are 95% confidence intervals.

Flow relations for shear and compression

The relationship between stress and tertiary creep rate for simple-shear and unconfined compression alone (Fig. 10) provides the basis for flow relations in these single component stress configurations. The tertiary creep stress exponent of n = 3.5 may be interpreted in the context of a flow relation with a cubic dependence on t_o and a stress-dependent level of enhancement, where Based on Eqn (5) and Figure 6, a suitable expression for in compression alone is

where E_c is the stress-dependent uniaxial compression strain- rate enhancement. The constant, e_c, is an enhancement scale factor for compression, and e_c − 6.3(MPa)^-1/2 when t ₀ has units of MPa. Similarly for shear alone,

The shear strain-rate enhancement, Es (Fig. 9), is determined by the shear enhancement scale factor, e_s −14.4(MPa)^-1/2. The ratio e_s: e_c −2.3 describes the relative magnitude of simple-shear and uniaxial-compression strain rates and is similar to values determined in other experimental studies (e.g. Reference Budd and JackaBudd and Jacka, 1989). Modelling using grain-scale anisotropic flow relations (Azuma and Goto-Reference Azuma and Goto-AzumaAzuma, 1996; Reference ThorsteinssonThorsteinsson, 2001) has shown that where crystal orientation fabrics compatible with simple shear (single maximum) and unconfined compression (small- circle girdle) exist, the higher levels of shear strain-rate enhancement in shear have a geometric origin. For an aggregate of grains, the mean resolved shear stress on the basal planes of the individual grains is higherfor simple shear on a single-maximum fabric where the mean orientation of c-axes is approximately normal to the plane of shear.

Flow relations for combined stress configurations

The stress configurations in natural ice masses are invariably a complex combination of compression, extension and shear components, which should all be considered when describing the flow. The laboratory deformation experiments in combined vertical compression and horizontal simple shear of Reference Li, Jacka and BuddLi and others (1996) and Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others (1999) led to the development of flow relations for combined simple shear and confined compression. These experiments showed that a transition in enhancement occurs when stress configurations vary from compression alone to shear alone. Our results add to this previous knowledge by refining values of E_c and E_s and describing their dependence on t ₀. Since E_c and E_s display a similar proportionality to t ₀ we propose a development of the flow relation of Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others (1999), on the basis that a similar dependence of E on t ₀ exists in combined stress configurations. From the current experimental stress configurations, we define an orthogonal coordinate system, where the shear stress, t , is applied in the x-direction, compression, a, in the vertical z-coordinate and the y-coordinate is normal to the xz-plane. In this reference frame, the deviatoric stress tensor, Sj, for simple shear and unconfined compression is

where S_xz = τ and S_zz = 2σ/3 are the shear and compression deviators and the magnitude of the corresponding octahedral shear stress is

In combined unconfined compression and simple shear (Eqn (8)), the proportions of the deviatoric components can be described by the shear fraction, λ_s (Reference Li, Jacka and BuddLi and others, 1996),

where for unconfined compression alone λ_s = 0 and in shear alone λ_s = 1.

Under similar conditions the shear-alone enhancement, E_s, is greater than that in compression alone, i.e. E_s /E_c = 2.3, which suggests that in combined stress configurations there may not be a direct relation between and t ₀ and the components of S_ij should be weighted by the appropriate component enhancements. Accordingly, we introduce the weighted anisotropic mean shear stress, T _o, of Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others (1999),

where a = E_s ^1/3 and β = E_c ^1/3 . Following Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others (1999), the corresponding shear () and compression () component strain rates in combined shear and compression are

The octahedral shear strain rate for tertiary creep, defined in terms of T _o, is (Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others, 1999)

By substitution of Eqns (9-11) into Eqn (14) the stress dependence of component strain-rate enhancement during tertiary creep is explicitly incorporated into the octahedral shear strain rate:

When λ_s = 0 or 1, Eqn (15) reduces to the corresponding relations for in compression or shear alone (Eqns (6) and (7)). In combined stress configurations where 0 < λ_s < 1 the octahedral shear strain-rate enhancement, E_o, is given by

By making the substitutions a = E_s ^1/3 and β = E_c ^1/3, and recalling that E_s =e_sT₀ ^1/2 and E_c = e_cT₀ ^1/2, Eqn (16) provides a description of enhancement factor E_o as a function of both stress configuration and magnitude. The family of curves in Figure 11 is based on this relation, and presents a hypothetical interpolation of enhancement in combined uniaxial compression and simple shear (Eqn (8)), for stresses from τ₀ = 0.025 to 0.40MPa. In developing their flow relation for combined shear and compression, Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others (1999) assumed a ratio of shear to compression enhancement factors of E_s /E_c ≈ 4. Our compilation of unconfined compression and simple-shear experimental data over a broader range of stresses provides a refinement of this ratio to E_s /E_c ≈ 2.3. Additionally, Eqn (16) builds on this previous work by providing a simple parameterization of the effect of stress magnitude, t_o, on enhancement that can be easily incorporated into other anisotropic flow relations that relate the strain-rate and deviatoric stress tensors. Examples include the phenomenological flow relation of Reference Warner, Jacka, Li, Budd, Hutter, Wang and BeerWarner and others (1999) and the grain-scale crystal orientation fabric based flow relations of Reference Azuma and Goto-AzumaAzuma and Goto-Azuma (1996), Reference Castelnau, Duval, Lebensohn and CanovaCastelnau and others (1996), Reference ThorsteinssonThorsteinsson (2001), Reference Gillet-Chaulet, Gagliardini, Meyssonnier, Montagnat and CastelnauGillet-Chaulet and others (2005) and Reference Placidi, Greve, Seddik and FariaPlacidi and others (2010).

Fig. 11. Tertiary creep octahedral shear strain-rate enhancement, Eo, as a function the shear fraction, λ _s, in combined unconfined compression and simple-shear stress configurations. As λ _s →0 the shear deviatoric stress S_xz →0, while as λ _s →1 the compression deviatoric stress S_zz → 0. The E _o–λ _s relationship (Eqn (16)) is plotted for a range of octahedral shear stresses, τ _o (MPa). The strain-rate enhancement for shear alone (λ _s =1) is ∼2.3 times the corresponding value in compression alone (λ _s = 0) for all valuesof τ _o.

Our results provide a clear illustration of the influence of stress configuration and magnitude on tertiary strain rates. Additional experiments over a broader range of temperatures and stresses are required to further describe flow in combined stress configurations relevant to the dynamics of large polar ice masses. Equation (15) has been developed from strain-rate data for stress configurations incorporating unconfined compression and/or simple shear: the limited availability of tertiary creep data from deformation experiments in other stress configurations currently precludes a broader description of E_o. We have recently completed a series of experiments that will assist with flow relation development by providing further data on individual component strain rates and their stress-dependent enhancement in combined shear and confined compression.

Low-stress tertiary creep behavior

Unlike isotropic minimum strain rates, which are associated with a transient stage of secondary creep at ˜1% total strain, steady-state tertiary creep rates are controlled by a dynamic balance between deformation processes, which increase the dislocation density, and recovery processes involving grain-boundary migration and the formation of strain-free nuclei. Other processes, including non-basal slip, grain-boundary sliding and diffusion, are considered to provide a minor contribution to recovery (Reference Montagnat, Durand and DuvalMontagnat and others, 2009; Reference Schulson and DuvalSchulson and Duval, 2009). Differences in the stress dependence of tertiary and isotropic minimum strain rates (Fig. 10) may be due to differences in ratecontrolling processes for the two creep regimes (Reference Schulson and DuvalSchulson and Duval, 2009). The convergence of tertiary and isotropic minimum strain rates at low stresses due to differences in their stress dependence (Fig. 10) raises the question of what, if any, changes to the stress dependence of steady-state tertiary creep occur at very low stresses. The availability of laboratory data to validate the low-stress creep behaviour is limited by the difficulties associated with conducting the long-running experiments necessary to determine very low tertiary creep rates.

The uniaxial compression experiments of Reference Jacka and LiJacka and Li (2000) at stresses from τ₀ =0.10 to 0.80MPa and temperatures of −5 to −45°C support the observed stress- dependent levels of strain-rate enhancement during steady- state tertiary creep and the resulting convergence of tertiary and minimum isotropic strain rates with decreasing stress. Reference Jacka and LiJacka and Li (2000) conclude this behaviour reflects a temperature-dependent shift in rate-controlling processes: at higher stresses and temperatures, migration recrystallization is dominant. As temperature and stress decrease, migration recrystallization becomes less influential, with other deformation modes, including rotation recrystallization and poly- gonization, controlling strain rates (Reference Montagnat, Durand and DuvalMontagnat and others, 2009; Reference Schulson and DuvalSchulson and Duval, 2009). In general, Reference Jacka and LiJacka and Li (2000) report lower enhancements than those determined from the current experiments, including E ≈ 1 at τ₀ = 0.10−20 MPa for experiments at −15 and −19°C. Two factors may have contributed to these low enhancement values. Firstly, the applied load was not periodically increased to offset the reduction in τ₀ resulting from sample barrelling. If no load adjustments are made during unconfined vertical compression, by 10% strain the strain rate will be reduced to ˜70% of the anticipated value. Secondly, in the experiments at τ₀ = 0.10 MPa the total accumulated strains may have been insufficient to develop a steady-state tertiary creep rate (e.g. total strains were <5% for experiments at −15 and −19°C). In comparison, an enhancement of E = 2.8 was determined from the current experiments at τ₀ = 0.20 MPa and −2°C when ε_o > 15% (Fig. 6). A single experiment at −45°C with τ_o = 0.20 MPa by Reference Jacka and LiJacka and Li (2000), in which two-stage experimental procedures were used to maintain a stable load, provides better agreement with the current results. The resulting enhancement of E =1.7 determined at ε₀ = 6% is an estimate of the minimum enhancement at τ₀ = 0.20MPa and −45°C, as the experiment was incomplete and the strain rate still accelerating when the results were published (Reference Jacka and LiJacka and Li, 2000). These observations suggest that while a low-stress reduction in enhancement to E ≈ 1 is possible, such a transition is likely to occur at a stress below the τ₀ ≈ 0.1 MPa suggested by the data of Reference Jacka and LiJacka and Li (2000) and therefore at a stress below which there are currently sufficient field and laboratory data to enable accurate quantification.

A range of experimental studies indicate a shift in the stress dependence of the creep power law to n < 3 at stresses of 0.1-0.3MPa (e.g. Reference SteinemannSteinemann, 1958b; Reference JackaJacka, 1984; Reference Pimienta and DuvalPimienta and Duval, 1987; Reference De La Chapelle, Milsch, Castelnau and DuvalDe La Chapelle and others, 1999; Reference Song, Cole and BakerSong and others, 2005). These data must be applied cautiously to steady-state tertiary creep as, with the exception of Reference SteinemannSteinemann (1958b), they relate to either isotropic minimum (secondary) strain rates or estimates of isotropic minimum strain rates determined by extrapolation of experimental data where total strains were significantly less than 1% (e.g. Reference Pimienta and DuvalPimienta and Duval, 1987). In uniaxial compression experiments where minimum creep rates were clearly determined, indications of a stress-dependent shift in creep behaviour are variable: Reference De La Chapelle, Milsch, Castelnau and DuvalDe La Chapelle and others (1999) report a transition to n = 1.8 ± 0.2 at axial stresses below ˜0.1 MPa (τ₀ < 0.05 MPa); the data of Reference JackaJacka (1984) indicate n ≈ 2 for τ₀ < 0.1 MPa, and Reference Song, Cole and BakerSong and others (2005) found n = 1 below 0.3 MPa (τ₀ = 0.14 MPa). Reference Song, Cole and BakerSong and others (2005) attribute their relatively high-stress transition in n to sample preparation techniques.

The derivation of robust estimates of n from field observations has proven difficult. As the stresses driving borehole closure are related to the overburden pressure, the Reference PatersonPaterson (1977) estimates of n = 3 from the closure of five polar boreholes generally correspond to stresses greater than 0.1 MPa, and the measured deformation is not steady- state, since primary, secondary and tertiary creep of the ice adjacent to the boreholes contributes to their closure. Reference Dahl-Jensen and GundestrupDahl-Jensen and Gundestrup (1987) obtained n = 2-3 from several different analyses of Dye-3 borehole tilt data; however, the derivation of robust estimates of n from inclinometer measurements is confounded by the requirement that all factors that influence the viscosity (temperature, polycrystalline anisotropy, impurities) be quantified in order to accurately determine the stress.

Despite a lack of conclusive laboratory and field data indicating a transition in the power-law stress dependence of tertiary creep, if one assumes that tertiary creep rates are always greater than or equal to the corresponding isotropic minimum values, then indications of a transition to n < 3 for isotropic minimum creep necessitate a transition to a similar or lower value of n during tertiary creep. Based on experimental indications of a transition to n < 3 for isotropic minimum creep rates at stresses below t_o = 0.1MPa, a transition from n = 3.5 to n < 3 during tertiary creep may occur at a similar or lower stress. Figure 9 indicates that for simple shear any transition in n must occur at t ₀ < 0.04 MPa.

In modelling flow in low deviatoric stress environments, such as ice divides, Reference Pettit and WaddingtonPettit and Waddington (2003) proposed a multicomponent flow relation with n = 1,3 and 5 terms. The linear term was found to be dominant at low stresses, and Reference Pettit and WaddingtonPettit and Waddington (2003) define the crossover stress as the deviatoric stress where the n = 1 and n = 3 components of the flow relation contribute equally to the total deformation. Using a combination of modelling and observations at Siple Dome, West Antarctica, Reference PettitPettit and others (2011) reported a crossover stress equivalent to τ₀= (1.5 × 10^-2) ± (2.0 ×10^-3) MPa.

Extrapolation of the experimental data for tertiary and isotropic minimum creep strain rates (Fig. 10) to low stresses allows calculation of a critical stress t₀ ^c at which the tertiary (n = 3.5) and isotropic minimum (n = 3) strain rates will be equivalent, resulting in E = 1. No physical significance is attached to τ₀ ^c ; it merely provides a reference for assessing differences in tertiary and isotropic minimum creep rates. For uniaxial compression t₀ ^c = (2.5 × 10^-2) ± (5.0 × 10^-3) MPa and τ₀ ^c= (4.8 × 10^-3) ± (2.0 × 10^-3) MPa for simple shear. It is noteworthy that the crossover stress of Reference PettitPettit and others (2011) is similar to t ₀ ^c = (2.5 × 10^-2) ± (5.0 × 10^-3)MPa for uniaxial compression, as t₀ ^c provides a lower bound for any transition from a tertiary creep regime with n = 3.5 to one with n < 3. While the occurrence of a transition in the tertiary creep regime is not excluded by the observed differences in n for tertiary and isotropic minimum creep, the mechanisms responsible plus the combined influence of factors including stress configuration, stress magnitude and temperature remain unclear. These uncertainties highlight the need for additional laboratory and field observations of low-stress steady-state tertiary creep behaviour. Conducting well-controlled laboratory experiments under the low-stress and -temperature conditions necessary to address these uncertainties is not trivial: depending upon experimental variables and techniques, 5 years or more may be required to obtain steady-state tertiary creep rates.