Introduction
Creep is the time-dependent deformation of solids under stress, where deformation rates due to the stresses are small enough so that failure does not occur. There is a distinct similarity of creep features in many different materials which has not yet been explained. The essential features exhibited by creep curves are: an instantaneous strain, a transient creep of decreasing rate, and a steady-state creep, approximately linear with time. At the present time there is not enough information on isolated processes to compile a comprehensive theory of creep behavior of’ snow as an aggregate.
Experimental Procedure
Three locations were used in the creep study of snow under low stresses. Tests were run at “Camp Century”, Greenland, for over two years, and tests were made over shorter periods at “Byrd” station and Amundsen-Scott South Pole station in the Antarctic. Undersnow rooms served as laboratories at all three locations. Temperature variations during the experimental period were ±0.5° C. for “Camp Century” and less at the Antarctic stations.
Two different experimental apparatuses were used for the present study. For the long-term tests, snow blocks and snow cylinders were loaded directly with dead weights, and the amount of deformation was measured by Starrett dial gauges, vertically mounted at 0°, 90°, 180°, and 270°, equidistant from the center of the sample. The four readings from each block were then averaged to determine total vertical deformation. Apparatus designed by Reference Butkovich and LandauerButkovich and Landauer (1960) was used for the short-term tests. Snow samples were cut to approximately 2 × 2 × 6 cm. and then compressed uniaxially at the center of an aluminum beam pivoted at one end with weights attached to the free end. A steel ball at the center of the beam transmitted the force through an aluminum plate frozen to the snow sample, and a mechanical Starrett dial gauge was mounted at the free end of the beam where deflections could be read to an accuracy of ±0.0002 cm. The entire apparatus consisted of twelve such units mounted on aluminum plates in groups of four.
Table I indicates basic information for the samples tested. The values given express the limits of the ranges studied.
Table I Summary of Conditions of Tests
Because of the small strains, it was assumed that the constant load did not appreciably affect the cross-sections, and the problem could be treated as one of constant stress. All samples were taken directly from pits and walls at the experimental sites. Particular care was taken to select snow samples which were more or less “homogeneous”, i.e. did not contain ice lenses or layers of depth-hoar crystals.
Results
Part A. Power function
Figures 1 and 2 show some typical strain versus time curves from “Camp Century”, South Pole and “Byrd” station. The higher curve for “Camp Century” in Figure 2 represents the test for snow of very low density and the change in slope is attributed to an increase in density of the sample at a high strain-rate.
Fig. 1. Creep strain versus time for all test sites
Fig. 2. Creep strain versus lime for all test sites, logarithmic scales
As can be seen from the figures, a power function
can be applied.Footnote ‡ Here ε 0 is an instantaneous extension, b and n are constants. Values for the constants b and n at each site are given in Table II.
Table II Values of Constants in the Transient Creep Law
Andrade’s equation (Reference Andrade1910, Reference Andrade1962) was also tried for some of the typical creep curves, this is of the form
where ε 0, β, k are constants. Exponents of
Part B. Rheological model
Figure 3 shows a typical creep curve for tests conducted over a long period (1 to 2 yr.) at “Camp Century”. Observations indicate an extended time period of from 100 to 350 days in the transient creep stage before constant viscous flow was attained. Later tests were conducted in the Antarctic during the summer of 1962–63 and a transient creep period of 90 days was recorded under the South Pole test conditions outlined in Table 1. Three variables (density, stress, and temperature) were observed to affect the length of time in transient creep. A low density snow would have more transient creep than a higher density snow, while a lower temperature or higher stress would reduce the transient creep. In metals they would increase the transient creep. Since the experiments run at the South Pole during the summer of 1961–62 were terminated after only ten days, it was necessary to extrapolate the data to a point of constant viscous flow in order to compare viscous strain-rate
Fig. 3. Creep strain versus time, and relaxation after removal of load, far “Camp Century” samples
Fig. 4. Compressive viscosity versus density for all test sites, logarithmic scale for viscosity. Results of confined compressive tests by Bader and Haefeli are added for comparison
The data as shown in Figure 3 show a similarity to Yosida’s results (Reference Yosida1956). The Theological model used is composed of a Maxwell unit and Voigt unit connected in series. An expression for total strain as represented by this model is
The first term σ/E 1 in the equation represents instantaneous elastic strain where σ is applied stress and E 1 is the elastic modulus. Constant strain-rate corresponding to viscous flow is indicated in the second term σt /η 1 where t is time and η 1 is compressive viscosity. This second term could also be defined as steady-state creep. The final part of the expression is the transient creep or decreasing strain-rate where E 2, is an elastic constant of the Voigt unit, and τ is the retardation time. It was assumed that in our test the strain rate is proportional to stress (Reference LandauerLandauer, 1955; Reference YosidaYosida and others, 1956).
Table III gives values for tests conducted at “Camp Century”, Greenland at a temperature of –22.5° C.
Table III Rheological Parameters Deduced from the Tests at “Camp Century”, Greenland
Figure 4 represents a composite plot of compressive viscosity versus density for the three locations. Values for the compressive viscosity were found by using the steady-state creep region where the strain-rate is constant. Reference Bader and KingeryBader (1963) gives an expression for the “densification viscosity factor” η c which the authors call “compressive viscosity” for the case of unconfined compression.Footnote *
where a and b are constants dependent upon site conditions and ρi is the density of ice. Applying Bader’s expression and constants for both South Pole and “Byrd” station (Table IV), two straight lines are obtained as shown in Figure 4.
TableIV Constants in Bader’s Expression for Densification Viscosity Factor
Discussion
Reference YosidaYosida (1956) applied classical rheology to his unconfined creep studies of snow. Our data, when treated in a similar manner, yielded good results. The composite graph of the compressive viscosity η 0 versus density p is shown in Figure 4. There is a rapid change in slope at the density 0.47 g. cm.−3 and again at 0.625 g. cm.−3 Yosida’s (Reference YosidaYosida and others, 1956, p. 1–14) data correspond to the general trend of the Figure 4 curves in the low density region (p < 0.47). For the density range 0.47 < p < 0.625, it was found that the compressive viscosity is nearly constant. This region will be referred to as the “plateau” region. Data from the South Pole indicate that when the density is greater than 0.625 g. cm.−3 the slope steepens to a higher value. The data from “Byrd” station and “Camp Century” do not include the high density range (ρ > 0.625), but it is believed that the trend will be similar to the South Pole curve. This is shown by the suggested curve.
In the “plateau” region there seems to be a rearrangement of snow grains without an appreciable change in viscosity. The activation energy for creep was calculated by applying the expression;
where η 0 and Q are constants characteristic of the material tested, R is the universal gas constant and T the absolute temperature for the snow at all three stations.
If the activation energy for the “plateau” region is calculated and the operation is extended through the density range 0.47 < ρ < 0.625, the activation energy Q is 7.2 kcal. Mole−1. This value is very low, only slightly higher than the energy needed to break one hydrogen bond. The indication here is that a single process dominates this “plateau” region.
Yosida reported activation energies for low-density snow (0.17 to 0.25 g. cm.−3), varying from 20.8 to 23.8 kcal. Mole−1 respectively. By extrapolating the viscosity curve to the low densities and calculating activation energies, partial agreement with Yosida’s values is found. There is actually a “varying” activation energy for the low densities (less than 0.45 g. cm.−3) and then again for densities greater than 0.66 g.cm.−3 up to ice. Reference LandauerLandauer (1955, Reference Landauer1957) obtained values of 13.4 and 14.0 kcal. Mole−1 for densities between 0.38 and 0.42 g. cm.−3. According to our results, a large change in activation energy occurs with a small change in density in the above density range. Reference Butkovich and LandauerButkovich and Landauer (1960) point out that the activation energy for ice (14.3 kcal. Mole−1) agrees with the previously calculated values for snow. It would appeal that coincidentally Landauer obtained these results for snow by working within the only density range in which there is an agreement in activation energy between snow and ice. Table V summarizes the values for activation energies found by the various investigators in different density ranges.
Table V Values for the Activation Energy for Steady-State Creep of Ice Found by Various Investigators
A possible explanation of the rapid change is that the activation energies in the high and low-density regions are the mean values for more than one process. However, this reasoning is questionable.
As has already been pointed out by Reference RamseierRamseier (1963), and substantiated by these data, there is more than one mechanism of densification operating over the whole density range. The entire process is a rather complicated one even Wit is separated into three distinct regions.
It is interesting to note that the air permeability curve (Reference RamseierRamseier, 1963, fig.3) shows a change in slope at the same density (0.625 g. cm.−3) as the viscosity curve. At ρ > 0.625 grains are deformed so that closure of air passages is effected to the extent that air can no Ionger pass freely around each grain. Air is trapped by the deforming grains to form air pockets, which in this stage of ice are evidenced by bubbles. This mechanism, deformation rather than packing and rearranging of grains, affects the creep as can be seen by the dramatic change in slope at density 0.625 g. cm.−3 (Fig. 4). The major mechanism governing the low-density region (Part I) is one of breaking bonds and “large” displacement in the structure.
The “plateau” region is of considerable interest in engineering problems where low stresses are applied. The desirability of having an almost constant strain-rate for a rather large part of the density profile would be evident in design to allow for settlement. Of course there is the question of how unconfined creep compares with creep of a block of snow in situ. The best viscosity values now available for natural conditions are obtained using Bader’s constants (Table IV). These values were plotted in Figure 4. It can be seen that the curves have approximately the same trend as the presently reported values up to the point where the “plateau” region appears and then again in the higher density range. This change of slope can be attributed to the fact that Bader’s analysis represented a smooth curve over the whole density range, while a more specific range of densities was used in this report.
Haefeli’s values (personal communication) for unconfined creep from the “Jarl Joset” station (mean annual temperature –27.8° C.) are also shown in Figure 4. Part 1 of his curve is in good agreement with Bader’s. The change in slope occurs at a density of 0.58 g. cm.−3, which is much higher than the ones for the other stations. This may be due to the fact that Haefeli’s were confined creep tests. Reference Mellor and HendricksonMellor and Hendrickson (in press) also conducted confined creep tests at “Byrd” and South Pole stations. The general trend of their data agrees with the authors’ but the scatter is large and the data are inconclusive for determination of the plateau region. Further confined creep tests of this type are needed.
Acknowledgements
This project was supported in part by the National Science Foundation, Office of Antarctic Programs. The authors would like to express their indebtedness to this group and also to Edward Oliver for his valuable assistance in the field work at the South Pole. Due to the length of experimental run at “Camp Century”, Greenland, it would be difficult to mention all the persons who provided assistance to the program, but special gratitude is given to Ernest Holt and Tom Long for their extra effort. Further appreciation is extended to Johannes Weertman for his critical review of the manuscript.
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