Introduction
The tensile strength of snow is an important quantity in snow mechanics. Destructive tensile fractures are habitually observed on the line where snow slabs break away from their anchoring zones (Reference Haefeli and ŌuraHaefeli, 1967; Reference PerlaPerla, [1975]).
The tensile strength of snow is a function of many variables including density, temperature, and snow type. In addition, any test introduces additional variables which must be considered including loading rate, sample size, boundary conditions at the places where the sample and testing equipment are in contact, and sample disturbance in mounting the specimens.
Experiments with small sample sizes using centrifugal tensile testers consistently show large scatter in the strength values and high mean strength values (Reference SommerfeldSommerfeld, 1974). In addition, centrifugal testers cannot be used in situ and therefore may subject the sample to damage upon insertion of the sample tubes. Jarring of the samples against the walls of the tester at fast acceleration rates is also possible in centrifugal tests.
The procedure used in the present paper to estimate the tensile strength of snow is intended to provide estimates for larger sample sizes closer to the expected sample size in natural slab avalanches. In addition, the tests were made in situ and the loads were applied slowly to prevent effects of jarring and sample damage. The boundary conditions at the places where the snow specimen and the testing equipment were in contact were such that the conditions expected prior to destructive tensile fracture observed in snow-slab release were approximately simulated.
Experimental Procedure
The experiments were done during the winters of 1971–72 and 1972–73 in the Cascade Mountains, U.S.A. The apparatus consisted of a series of large tables set out on a horizontal snow surface to collect samples of snow. The tables had removable, wooden sides such that the samples were collected in a box on top of the tables. Following collection of a sample, the top was carefully trimmed off and the wooden sides were removed to provide uniformly shaped and sized samples. The surface of the tables was partly rough and partly smooth. It was hoped that destructive tensile fracture would occur on the line between the rough and smooth sections as the table was slowly tilted by lifting one end resulting in slow loading by body forces. This proved to be the case for each test. The fracture surfaces were, in general, smooth and perpendicular to the table surface.
The stress field in the samples in the present tests is not homogeneous. The tests were designed to simulate conditions of tensile failure in slabs whose resistance to sustaining basal shear stress is diminished. There will, therefore, be small differences when the present data are compared to uniaxial test results. However, a smaller effect is to be expected when comparing the results to field estimates.
Figure 1 shows a schematic drawing of the tilting table. The rough portion of the table surface was made by nailing wooden strips to the table surface near the end of the table that was to be tilted. The smooth portion of the table surface was achieved by constructing a cart on wheels with a rough surface on the top of the cart. Friction experiments with zero tensile force showed that this arrangement produced a coefficient of friction that was very small in comparison with other errors.
Fig. 1. Schematic drawing of tilting-table apparatus used in tensile tests.
The dimensions of the samples were 0.91 m wide by 1.83 m long with sample heights on the order of 0.3 m. The samples were shaped (notched) in the vicinity of the interface between the rough and smooth sections of the table (Fig. 1). Typical fracture areas were on the order of 0.12 m2 in cross-section.
Parameters recorded included average density, average temperature, snow type, age, and time to failure. The greatest source of measurement error in these experiments was in the estimation of the average snow density on the tables. In order to minimize this error a large number of density samples were taken with a cylindrical sample cutter 0.15 m in diameter through the depth of the sample and with a small sample cutter with a volume of 200 cm3 (0.2 × 10−3 m3).
The angle of tilt of the table when fracture was achieved was measured with an inclinometer to an accuracy much greater than required in relation to other errors. Angles of tilt at failure ranged from 10° for snow with the lowest density to 45° for higher-density snow. The tables were tilted slowly to avoid jarring and disturbance from in-situ conditions.
Several experiments were conducted to test the effect of notch geometry at the interface between the smooth and rough parts of the table on the results. The results showed no discernible notch sensitivity for notch shapes ranging from very thin slits to rounded circular forms.
Table I summarizes the tensile strength estimates as functions of the measured field parameters.
Table I. Tensile Strength of Snow Versus Measured Field Parameters
Discussion of Tensile-Strength Results
The most important easily measurable parameter to relate to tensile strength of snow is snow density (Reference SommerfeldSommerfeld, 1973). Figure 2 shows a comparison of the present estimates with the scatter band of the only other in-situ estimates which were given by Reference PerlaPerla (1969). Perla’s experiments were made by rapidly undercutting cantilever beams of snow in pit walls, presuming tensile failure in the top fibre. Perla’s experiments differ from the present experiments in some important ways: (1) the fracture patterns observed in the beam tests indicate that fracture was not always in tension and may have included shear and bending effects, (2) the beam tests were done at faster rates than the present experiments and the possibility of jarring and sample disturbance is higher, (3) the effective sample fracture area in the cantilever beam experiments was on the order of 10% of that for the present tests.
Fig. 2. Comparison of tensile-strength estimates from present experiments with the limits of the in-situ data given by Perla (1969).
If one accepts the definition of brittle fracture as fracture without significant prior plastic deformation (Reference Erdogan and LiebowitzErdogan, 1968), then it is not possible, strictly speaking, to classify the present experiments as brittle or ductile since no deformation measurements were made prior to fracture. Since the experiments were all done at temperatures near the melting point, some ductility is expected. In addition, the samples were loaded for periods of time on the order of minutes prior to fracture will most likely have given time for plastic deformation.
If one ignores, for the moment, effects of ductility and inhomogeneity of stress conditions, comparison of the present data with the in-situ data of Perla and the non-in-situ data from centrifugal tests (Reference Keeler and WeeksKeeler and Weeks, 1968; Reference SommerfeldSommerfeld, 1974) leads to the conclusion that increased sample size results in lower mean values of strength and less scatter as a function of density. These results are consistent with the predictions of Weibull-like statistics (Reference Freudenthal and LiebowitzFreudenthal, 1968) as proposed by Reference SommerfeldSommerfeld (1973) for application to brittle fracture of snow. In fact, the minimum strength values obtained in the present experiments lie close to the values for large sample sizes predicted by Reference SommerfeldSommerfeld (1974) by applying Weibull statistics to centrifugal-test data. This is shown in Figure 5.
Fig. 5. Comparison of the limits of the data from the present tilting-table tensile experiments with the estimates of tensile strength for large sample sizes by Sommerfeld (1974) and with estimates of shear strength from avalanche fracture lines compiled by Perla (1977). The solid lines denote the limits of the present experiments. The shaded portion denotes the shear-strength estimates by Perla (1977) . The heavy line (
However, use of Weibull statistics presumes brittle fracture by a specific mechanism. Size effects are expected in both brittle fracture and ductile fracture (Reference Freudenthal and LiebowitzFreudenthal, 1968). For the present experiments not enough tests are available to warrant use of such statistics and, in addition, the mechanism of fracture is not known, so that Weibull statistics, properly speaking, cannot be applied in this case.
Attempts to Measure Shear Strength in situ Utilizing Large Sample Sizes
Estimates for shear strength were attempted during the winter 1972–73 in the Cascade Mountains. The basic method of sample collection was similar to that for the tensile tests. Figure 3 depicts the apparatus. The central portion of the sample consisted of a cart on wheels. The cart was flanked by two fixed portions of the sample. All three portions where the sample contacted the apparatus had rough surfaces.
Fig. 3. Schematic drawing of the apparatus used in the experiments attempting shear-strength estimates.
The basic procedure consisted of slowly pulling the cart by a hand-cranked winch until failure. The table was in a horizontal position during the entire experiment. A maximum recording dynamometer was attached into the cable connected to the winch. Parameters studied during these tests were the same as described for the tensile tests, including sensitivity to notch geometry, which was found to have little effect. The loading was applied over a time scale of one to two minutes.
Figure 4 depicts a typical fracture pattern. The four cracks were measured to be near 45° to the direction of pull in each case. These fracture patterns indicate that the samples failed in tension. It is difficult to define failure strengths in such experiments and perhaps it is not worthwhile in view of the complicated stress conditions expected to be associated with such tests.
Fig. 4. Pattern of cracks observed in experiments attempting shear-strength estimates. The pattern indicates tensile fracture.
Interpretation of Observed Fracture Patterns
The fracture observed in the tilting-table tensile tests were all relatively smooth and perpendicular to the plane of the tilting table. The stress conditions in such a test would be expected to produce maximum principal tensile stresses approximately parallel to the table. Fracture in a direction perpendicular to the important tensile stress is often observed in geologic materials (Reference JaegerJaeger, 1962) and is observed in the release of snow-slab avalanches where a catastrophic tensile fracture perpendicular to the shear failure plane is observed (Reference Perla and LaChapellePerla and LaChapelle, 1970).
It is more difficult to analyse the tensile fracture patterns observed in the attempted shear tests. However, it is perhaps not surprising that the fractures were in tension in view of the general difficulty of generating shear fractures in isotropic, homogeneous samples of materials.
Magnitude of Tensile Strength Measurements with Respect to Avalanche Release
Reference MellorMellor ([1975]) pointed out that approximate values of the shear strength of snow might be made by analysis of depth and density profiles at avalanche fracture lines. Reference PerlaPerla (1977) vised this procedure to estimate shear-stress failure levels versus snow density at the weak shear failure plane for 69 slab avalanches. These are the only easily obtainable data in regard to avalanche-release failure levels. It is not possible to relate the shear strength to the tensile strength of snow at present, because the failure surface is unknown. Common failure theories for metals predict that the shear strength is one-half to two-thirds of the tensile strength. Granular materials such as snow, being much stronger in compression than in tension, might, however, be expected to have a shear strength which approaches the tensile strength due to possible widening of the failure envelope.
Figure 5 shows a comparison of the limits of the estimates from the present tensile experiments with the limits of the shear-strength estimates from fracture-line profiles as a function of density. Most of the estimates of shear strength from fracture-line profiles represent failures in planes which are especially weak in shear due to recrystallization under strong temperature gradients or other effects. Therefore, Figure 5 may indicate some aspects of the importance of strength anisotropy in the avalanche failure-strength problem. If failure strengths could be compared for homogeneous samples, presumably the tensile strength would be approximately equal to the shear strength. On the other hand, the comparison in Figure 5 indicates values of tensile strength somewhat in excess of estimated shear strength values for field conditions, perhaps indicating that the anisotropy observed at avalanche fracture lines (i.e. a noticeably weak shear failure plane under the slab) is important. However, the very approximate nature of the comparison must be kept in mind since there are differences in sample size, rate of strain, and other factors in the above comparisons. It is likely, for example, that the present tests are not of large enough sample size to be regarded as minimum estimates (Reference SommerfeldSommerfeld, 1974).
Summary
In-situ estimates of the tensile strength of snow have been given by a new method utilizing large sample sizes. The data from the experiments show the effect of large sample sizes on tensile strength of snow indicating less scatter in the results as a function of density and lower mean values than previous in-situ tests. The data support (but do not prove) the idea that Weibull-like statistics apply to fracture of snow.
Since the samples were loaded slowly and the measured sample temperatures were near 0°C, the tests will probably have had plastic deformation associated with them. Most avalanche fractures occur in snow packs with bed surfaces of temperatures greater than −10°C (Reference Perlaperla, 1977). From the standpoint of avalanche release, the rate at which tensile loading occurs near fracture lines is not presently known, however, so that it cannot be stated whether the slow loading employed detracts from the usefulness of the data. The advantage of slow loading in preventing jarring may well override any such details.
Acknowledgements
The field experiments for this paper were done as part of my doctoral thesis at the University of Washington. The basic idea for the experiments is due to Professors C. B. Brown and R. J. Evans. I am grateful to them for many useful discussions in regard to this work. The field work was funded by the Washington State Highway Department under Contract Y-1301.
I am grateful to Dr K. Høeg of the Norwegian Geotechnical Institute for helpful suggestions in writing the manuscript.
