## Creep of finegrained ice

The rheological behaviour of materials can be expressed in a general flow law of the form

dp IRT

where A is a material-dependent parameter, d is grain size, p is the grain-size exponent, s is differential stress, Q is the activation energy for creep, R is the gas constant and T is absolute temperature.

Ambient-pressure experiments on ice samples with uniform grain sizes of 3 |lm to 0.2 mm (hereafter referred to as 'finegrained') unequivocally reveal the existence of three creep regimes, shown schematically in Fig. 60.1, each with characteristic values of n, p and Q: dislocation creep, GBS-accommodated basal slip (i.e. GBS-limited) creep and basal slip-accommodated GBS (i.e. basal slip-limited) creep (Goldsby & Kohlstedt, 1997, 2001). A fourth creep mechanism, diffusion creep, should dominate the flow of ice at the lowest stresses but has not yet been accessed in the laboratory (Goldsby & Kohlstedt, 2001).

At high stresses (>1MPa), dislocation creep of fine-grained ice is characterized by a value of n ~ 4 and is independent of grain size (i.e. p = 0, see fig. 2 of Goldsby & Kohlstedt, 2001). Dislocation creep data for fine-grained ice are in near-perfect agreement

Ml O

GBS-limited | |||||||||||||||||||||||||||

creep n = 1.8 | |||||||||||||||||||||||||||

basal slip-limited | |||||||||||||||||||||||||||

creep n = 2.4 |
dislocation creep n = 4.0 | ||||||||||||||||||||||||||

diffusional flow | |||||||||||||||||||||||||||

n = 1 |
Figure 60.1 Schematic diagram depicting four ice-creep regimes: Dislocation creep, characterized by n = 4.0 and p = 0, GBS-limited basal-slip creep, with n = 1.8 and p = 1.4, basal-slip-limited GBS creep, with n = 2.4 and p = 0, and diffusion creep, with n = 1 and p = 2 or 3, depending on whether volume or grain-boundary diffusion, respectively, is rate-limiting. Creep data for the n = 2.4 creep regime for fine-grained ice are in excellent agreement with creep data for single crystals of ice oriented for basal slip (e.g. Wakahama, 1967). The heavy solid line represents the composite flow behaviour of ice given by Equation (3). with results of an exhaustive study of dislocation creep of coarser grained (>0.25 mm) samples deformed at elevated confining pressures (e.g. Durham etal., 1992; 1997). Microcracking, sometimes invoked to explain values of n > 3 in ambient pressure tests, is suppressed in elevated pressure experiments. Dislocation creep data for fine-grained ice are also in very good agreement with results of experiments on coarse-grained (>1mm) ice samples deformed at ambient pressure and stresses above ca. 1 MPa (e.g. Steinemann, 1958a; Barnes etal., 1971); at a given stress and temperature, strain rates determined at elevated pressure vary from strain rates measured at ambient pressure by less than 10% (see fig. 7 of Goldsby & Kohlstedt, 2001). The activation energy in the n = 4 creep regime, below the onset of pre-melting at ca. 255 K (see Goldsby & Kohlstedt, 2001), is equivalent to that for volume diffusion of hydrogen and of oxygen (ca. 60kJ mol-'). The value of n = 4 and the equivalence of the activation energy for creep with that for diffusion of hydrogen and oxygen are consistent with models of climb-limited dislocation creep (e.g. Weertman, 1968). With decreasing stress, a transition occurs from the dislocation creep regime to the GBS-limited creep regime in which compatible deformation occurs via dislocation slip on primarily the basal-slip system combined with GBS. Although GBS contributes significantly to the strain rate in this regime, the strain rate is probably dominated by basal dislocation slip and rate-limited by GBS (Goldsby & Kohlstedt, 2001). The GBS-limited creep regime is characterized by n = 1.8. Unlike dislocation creep, the strain rate due to GBS-limited flow depends strongly on grain size, with p = 1.4.
*This value of A for dislocation creep is revised from the original value in Goldsby & Kohlstedt (2001), providing a better fit to the combined data set for dislocation creep from high pressure and ambient pressure tests at T < 258 K. *This value of A for dislocation creep is revised from the original value in Goldsby & Kohlstedt (2001), providing a better fit to the combined data set for dislocation creep from high pressure and ambient pressure tests at T < 258 K. As stress is decreased further, a transition occurs to a creep regime in which the creep rate is probably dominated by GBS but rate-limited by basal slip. The basal slip-limited creep regime is characterized by a value of n = 2.4 and the strain rate due to this mechanism is independent of grain size. Both the value of n and the absolute magnitude of the strain rate at a given stress in this regime are in excellent agreement with data for single crystals of ice oriented for basal slip (e.g. Wakahama, 1967), as shown in fig. 2a of Goldsby & Kohlstedt (2001). At the lowest stresses, a transition is expected to the diffusion creep regime (Mukherjee, 1971), which is characterized by a value of n = 1 and a strong dependence of strain rate on grain size (with p = 2 or 3 depending on whether volume diffusion or grain-boundary diffusion, respectively, limits the creep rate). Using a value for the unknown parameter in the diffusion-creep equation, the grain-boundary diffusion coefficient, constrained by experiments on fine-grained ice, Goldsby & Kohlstedt (2001) concluded that investigation of diffusion creep in the laboratory would require the maintenance of submicron grain sizes at T > 248 K. This requirement may render diffusion creep 'inaccessible' in the laboratory. Goldsby & Kohlstedt (2001) proposed a composite flow law for ice that includes contributions to the overall creep rate from the four mechanisms described above Each term in the composite flow law is a flow law of the form of Equation (2), where the subscript 'diff' denotes diffusion creep, 'basal' denotes basal slip-limited creep, 'GBS' denotes GBS-limited creep, and 'disl' denotes dislocation creep. Using experimentally determined values of A, n, p and Q for dislocation creep, GBS-limited creep and basal-slip-limited creep, along with estimates of the diffusion-creep rate, Goldsby & Kohlstedt (2001) extrapolated creep data for fine-grained ice via Equation (3) to larger grain sizes (>1mm) and to temperatures up to 273 K. Such extrapolations indicate that either GBS-limited creep or dislocation creep is the rate-limiting creep mechanism for ice over a wide range of conditions of glaciological interest, i.e. grain sizes > 1mm, stresses higher than ca. 0.0001 MPa and temperatures >220K (Goldsby & Kohlstedt, 2001). Therefore, a simplified form of the composite flow law can be used for the purposes of comparison with previous laboratory data and for glacier and ice-sheet modelling The flow-law parameters A, n, p and Q for the GBS-limited creep and dislocation creep regimes are given in Table 60.1. |

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