The creep of ice has been studied for many years both in the laboratory and in the field. However, although considerable data are available, only a small fraction of these are useful for determining the mechanism of the creep process and relatively little effort has been made to interpret these data in terms of modern ideas of the plastic deformation of solids.
Recently Reference Weertman and KingeryWeertman (1963) has discussed the creep data for ice single crystals oriented for basal slip due to Reference SteinemannSteinemann (1954) and Reference Butkovich and LandauerButkovitch and Landauer (1958) in terms of a dislocation damping mechanism due to Reference SchoeckSchoeck (1956) and Reference EshelbyEshelby (1961), and of a dislocation climb mechanism due to Reference WeertmanWeertman (1957 [a]). Both mechanisms predict creep rates which agree within an order of magnitude with the experimental data but Weertman favors the former since it predicts the experimentally observed stress dependence.
At a given stress, the creep of polycrystalline ice is 102 to 103 times slower than that of single crystals oriented for basal slip and Weertman suggests that this is due to the resolved shear stress on the basal plane of any grain in a polycrystalline aggregate being smaller than the applied stress. However, the observation of Butkovitch and Landauer that crystals oriented to suppress basal slip gave creep rates similar to those of polycrystals for similar stresses at the same temperature is so striking an analogy to the observations of Reference GilmanGilman (1956) on high temperature deformation of zinc that an alternative suggestion is worth examination. This note seeks to show that creep of polycrystalline ice near the melting point is controlled by non-basal slip in the crystals that are oriented with their basal planes parallel to the stress axis.
The concept of non-basal slip as an important deformation mode in ice receives support from the recent experimental observation of Reference MugurumaMuguruma (1963) and Reference Muguruma and HigashiMuguruma and Higashi (1963) showing that slip can occur on {1122} and {1012} planes as well as on the basal plane.
In the case of zinc, the steady-state rate for prismatic slip in single crystals was found by Reference GilmanGilman (1956) to be related to stress and temperature as:
where
For ice, the temperature dependence of the creep rate of crystals oriented for non-basal slip has not been determined but the reported value of n = 2.96 is in agreement with the observations on zinc and cadmium. Observations on polycrystalline ice are conflicting and it is difficult to assign a definite value for n and Q (see, for example, Reference GlenGlen, 1958, Reference Glen and Kingery1963). It has been pointed out by Flinn in a private communication that the ratio of the activation energy for non-basal slip to the melting temperature is fairly constant for the hexagonal metals at 55 cal./°K. in contrast to the ratio of 39 cal./°K. for the ratio of the activation energy for dislocation climb to melting temperature which is to be expected from the idengification of the activation energy for dislocation climb with that for self diffusion. If we extend this correlation to ice, then we predict a value of 10,600 cal./mole for a dislocation climb process, similar to that suggested by Reference Weertman and KingeryWeertman (1963), and a value of 15,000 cal./mole for a non-basal slip process, similar to that suggested by Butkovitch and Landauer.
Two mechanisms have been proposed for prismatic slip in hexagonal metals. The first, due to Reference WeertmanWeertman (1957[b]) is based on overcoming a large Peierls stress on the prismatic plane, while the second, due to Reference Friedel, Rassweiler and GrubeFriedel (1959) is based on thermally activated cross-slip from the basal plane into the prismatic plane. The former yields a stress exponent of 2.5 at low stresses while the latter gives an exponent of 2. It is possible to calculate activation energies for the two mechanisms but both depend on the assumed width of the dislocations since this determines the Peierls stress in the former case and the constriction energy in the latter case.
One way to use Weertman’s model for the case of ice is to take the value of Q as predicted empirically and see whether a reasonable value of the Peierls stress emerges after substitution in his expression for Q which is:
where d is the distance between Peierls hills, b is the Burgers vector, μ is the shear modulus and τ is the Peierls stress. Using μ = 3×1010 dyne/cm.2, a = 4.5Å, c = 7.35Å, the calculated value of τ is 3×1010 dyne/cm.2 which is similar to that calculated for zinc from the experimental value of Q.
Assuming then that the predicted Q is reasonable, we can calculate the creep rate from Weertman’s equation which simplifies to:
where all the quantities are in c.g.s. units. This predicts a rate of 10−7/sec. at σ = 1.kg./cm.2 at −5°C. in reasonable agreement with the experimental value of 10−8/sec.
In Friedel’s model, the activation energy consists of two terms as
where U c is the constriction energy and x is the ratio of the energy gained per unit length in dissociation to the line tension of undissociated dislocation, about 10−2 for reasonable values. If the dislocation is undissociated, U c can be neglected and the second term is dominant while if the dislocation is dissociated the reverse is true. Similar calculations to those of Reference GilmanGilman (1956) indicate that the dislocations in ice are probably undissociated and thus use of the second term gives Q ≃ 14,000 cal./mole for σ = 1kg./cm.2 in reasonable agreement with the empirically predicted value. The creep rate can then be calculated from Friedel’s equation which simplifies to a similar expression to that of equation (3), namely:
where all the quantities are again in c.g.s. units. For σ = 1.kg./cm.2 and −5°C., this also gives a creep rate of 10−7/sec.
Thus either mechanism can describe the observed creep behavior of ice single crystals oriented for non-basal slip although possibly more weight can be given to Friedel’s mechanism in view of the recent observation of cross-slip in ice by Muguruma.
In the case of the zinc polycrystals used by Gilman, the stress required to produce a given rate at a given temperature is about 30 per cent lower than that for crystals oriented for prismatic slip. Examination of the data of Butkovitch and Landauer for ice indicates a similar but smaller trend. This stress difference arises because only a fraction of the grains in the aggregate are oriented for non-basal slip and thus initially the grains oriented for basal slip will deform to accommodate the applied load. However, as deformation proceeds, the load on the less favorably oriented grains increases until the local stress is sufficient to activate non-basal slip in these grains which then control the rate of deformation. Because of local inhomogeneities of strain, minor deformation modes, e.g. folding and boundary migration, occur as described by Reference Gold and KingeryGold (1963). From his observations, he ascribes a minor role to non-basal slip in contrast to the viewpoint expressed here.
From the above discussion, the creep of polycrystalline ice near the melting point is considered to be controlled by non-basal slip and hence the creep rate can be expressed by an equation similar to that of equation (3) or (5), depending on the exact mechanism of the slip process, with a factor in the pre-exponential term to allow for the distribution of grain orientations, i.e. a factor dependent on texture. Further, by analogy with the results on hexagonal metals, we expect a change in the rate-controlling process to that of dislocation climb at temperatures around −65°C. to −100°C. with a change in stress exponent to around 4.5 and a change in activation energy to around 10,500 cal./mole. Clearly further careful experimental work is desirable to check these ideas.
I wish to thank Professor J. Weertman for his interest and encouragement.
