On Mechanics of the Hard Slab Avalanche
T.E. Lang,* R.L. Brown*
Various factors contribute to the recognized uncertainty over the mechanism of hard slab snow avalanche release. Perhaps the greatest difficulty is the hazard of field observation of other than the crown region of an avalanche sensitive slope Additionally, down-slope pre-avalanche material and geometry conditions are largely obliterated following the avalanche. The up-slope or crown region post-avalanche geometry remains in tact, and has been monitored and studied in some detail. Theories of release have been formulated assuming initial disturbance in the crown region, as by a tensile stress fracture, and subsequent propagation down-slope(1).
Snow, not being an easy material to characterize theologically, has complex load-deformation properties and exhibits unique structure changes under certain thermal history conditions. In general, load, deformation, microstructure, and thermal characteristics of snow in-situ on slopes have not been systematically measured, so that quanitization of factors relevant to subsurface conditions are not known. However, recognizing the fact that physical changes occur in the interior of snow pack, release theories have been formulated based upon some type of basal layer inhomogenity. It is assumed that material transformation or a form of inclusion produces a weakened state or an unstable structural configuration Either from local collapse or a shear failure, the release ensues (2,3,4). Recent work which supports this concept of a weak sublayer are the theoretical study of the stress state associated with this geometry (5,6), and an order-of-magnitude evaluation of the feasibility of a buckling mechanism contributing to the enhancement of slope failure (7). These studies relate to subsurface and toe region influence upon the release question.
If, indeed, these mechanisms exist and affect release, then experimental and modeling techniques will have to be developed to aid in evaluating their importance. Based upon evidence now known, it is reasonable to assume the existence of a weakened basal layer condition. But there may be a number of disturbance types or imperfections which induce the triggering of the avalanche as the slope, by some process or other, reaches a critical stability state. If monitoring and control of the avalanche sensitivity of a slope is desired, then the important question is what physical changes occur early enough and with sufficient magnitude to reliably serve as a measure of slope stability. One possible macroscopic mechanism that may be detectable is local buckling of the slope. Evidence of long term large- amplitude buckling of snow pack is well documented, and the question arises whether buckling is a primary or secondary mechanism in avalanche release? Lacking conclusive experimental evidence of the importance of buckling, the concept is further explored in the remainder of this paper.
Material Representation for Buckling Analysis
The formulation of a possible buckling state is strongly dependent upon an adequate material characterization. To date an extensive variation exists in the constitutive properties used to study snow response. Most analyses are based upon linear constitutive equations, and time dependence expressed by a deformation or strain rate term (viscous response). In setting up a buckling model it would appear that refinement in the constitutive law to account for more than one rate dependence can be treated. In constitutive law modeling to date, a linear viscoelastic model of low density snow has been reported by Shinojima (8); however, in the absence of stress relaxation considerations, the model is based upon long-term fluid behavior. Results by Yosida (9) and in tests conducted by the authors (10), long term solid material residual is observed, in that, complete stress relaxation under constant deformation does not occur. An additional complication reported by Shinojima (8) is that the linear form of the constitutive equations is different for each type of loading investigated, which included simple tension, compression and torsion.. Thus, different material coefficients should be used depending upon local stress conditions. However, this form of material non-linearity should not be a primary factor in formulating a buckling criterion. The existence of a weakened sublayer, which is generally recognized as a necessary condition for slope instability, results in incomplete stress transfer to the slope bed surface and a transmittal and intensification of bearing stress downslope. Thus, the toe region material is in a state of compression, which simplifies the requirements on the constitutive representation.
The non-linearity noted by Shinojima (8) in transition from a compression to a tension state is reflected in the value of Poisson's ratio. In tension the Poissonic effect approaches that of an equivoluminal material, whereas in compression the Poissonic effect is small. This difference in material behavior under
different types of loading is attributable to the skeleton crystal structure of snow, in which both volumetric and distortional deformation mechanisms act. This is markedly different from typical viscoelastic modelling assumptions, but should be accounted for in setting up a viscoelastic model of snow.
What is perhaps the greatest impediment to a simple constitutive representation of snow is the fact that snow behaves strongly non-linearily to changes in deformation rates, loading sequences, etc. Yosida (9) indicates a strong nonlinear relationship between normal stress and low strain rates in simple compression tests of snow columns. Application to analysis of buckling can be handled by equivalent linearization of the constitutive model in the standard method of treating material non-linearity.
The behavior of snow is complicated by its dependence on a number of items, which includes temperature, density, and state of metamorphism. The state of metamorphism, as indicated by Yosida (9) can be characterized in terms of the thermal history and stress history of the material. These considerations therefore make the complete thermomechanical characterization of snow an extremely difficult task to undertake. However, this approach of characterizing snow is probably not necessary for making a comprehensive analysis of the problem of buckle mode growth. It is quite possible, as indicated by Yosida (9), that a large portion of the snow slab may be metamorphically stabilized during the months of January through March, and that the time-wise variation of the material properties may be negligible. If this is the case, the material aging characteristics and thermal history effects may be neglected in formulating the material constitutive equations, which must necessarily be non-linear. However, since the stress distribution in the slab is compressive, and the range of stress through the depth of the slab may be restricted, the use of equivalent linear constitutive equations can be considered a valid simplication. However, more research needs to be done to verify if this can be done. Some questions pertaining to this which must be answered are: first, the extent to which one simplified constitutive equation can be utilized to represent the entire slab (i.e., the effect of density variation and the percentage of the slab which does stabilize metamorphically), and, second, the correlation between stabilization of metamorphosis and macroscopic material properties.
In summary, the key to the analytic treatment of the buckling question is a refined model of the constitutive representation coupled with simplifying assumptions on the range of parameters based upon the physical conditions of the slab buckling phenomenon.
Physical Characteristics of Snow Slab Buckling
Two buckling geometry's can occur. One is buckling of the surface layer of the snow pack while supported by a bed surface cushion. This requires either an interstitual weak layer (as from water percolation or material stratification), or a metamorphized basal layer (as from formation of depth hoar). Perla (5) determined from examination of a number of post-avalanche slopes that in 65% of the cases depth hoar was in evidence. Admitting the mechanism of long term buckling, the wave shape of a typical buckling mode induces local regions of bearing stress intensification on the basal layer of depth hoar. This overstress enhances the brittle fracture and collapse of the depth hoar matrix, and, thus, is a plausible mechanism as an initial triggering source.
The second buckling geometry is the formation of a buckling pattern of the entire slab, which implies cavity formation at the bed surface. Two alternatives exist here, that either the cavity exists and buckling follows, or that the tendency for buckling produces the cavity. Whichever is the case, the formation of a buckle lobe in the toe region produces a geometric and stress intensification configuration that enhances the formation of a slip plane (Figure 1). Alternately, the feasibility of a buckling mechanism "locking" a slope must also be examined.
To examine the question of whether or not buckle formation is physically possible in snow slabs, snow columns were tested in compression, and the material coefficients determined, were used in a buckling analysis. Snow columns of nominal length 20 cm, and specific weight .39 gm/cm3 were tested at -10 degrees C at constant deformation rates up to 0.005 cm/min. The load-deformation data was fit by a linear three-element viscoelastic solid model and a buckling analysis procedure was followed as developed in Reference (7). Results of the computations are shown in Figure 2. The interpretation is that for a given length of bed-surface imperfection having an initial amplitude 0.05 of its length, the curve shown is the boundary between growth and subsidence of the imperfection. The abscissa is the factor indicating the number of equivalent lengths of imperfection that must be bearing onto the imperfection zone to yield a corresponding rise time for an order-of-magnitude change in the amplitude of the imperfection. Thus, for an imperfection of length, l, snow of equivalent length 4l must bear onto the imperfection zone in order that the amplitude of imperfection increase by a factor of ten in 103 hrs. or approximately 41 days. Thus, even though the snow specific weight is high and the test temperature is low for mid-alpine snow pack (both factors, if adjusted accordingly, decrease the time for amplitude growth), a reasonable estimate of a buckling mechanism is obtained.
To further define whether or not sub-surface imperfections can form, a snow slope in the Bridger mountain range north of Bozeman, Montana having a history of avalanche activity, was selected. A 40 meter long trench was dug along the nominal 40 degree slope approximately 1/3 distance in from the left flank of the snowpack, which terminates into tree and rock outcrops on both flanks and at the crown. Void imperfections were found, as depicted in Figure 3, which encompassed 40% of the 40 meter length. All voids were easily distinguishable, the largest having an amplitude of approximately 12 cm, and all voids extended under the snowpack indicating the exposed section was probably typical. Approximately 5 meters from the crown region tree outcrop a crack 20 meters in length and 0.3 meters in maximum separation ran parallel to the outcrop. The existence of this crack indicates that the particular slope was in a state of glide. However, the significant fact is that the void formation mechanism was more pronounced than snowpack settlement, which would cause subsidence of the voids. Thus, it is probable that slope creep rate relative to snowpack settlement rate is an important parameter on whether voids form or not. Once voids develop any weakened subsurface condition that would result in incomplete shear stress transfer to the bed surface would reinforce void growth.
Final consideration is here given to a geometric profile described by Perla (5). He indicates that the flank fracture profile of an avalanche zone has the shape of a sawtooth in a majority of cases. The sawtooth slants downslope, and conforms exactly to what would be predicted if a buckle wave with a downslope sagging central region were present immediately preceding avalanche release. Although the flank regions are influenced strongly by edge intrusions of rocks, trees etc., if regular dimensions can be ascribed to the sawtooth pattern, this might relate directly to dominant buckling mode dimensions. Perla reports no data of this type, and the authors find none available in the literature.
Based upon the initial evidence concerning slab buckling, it appears that the possibility of a buckling state developing in avalanche sensitive slopes must be considered. Using a simplified analytical model and nominal material properties, the times computed for buckle mode growth are at least of the same order as times
associated with the avalanche phenomenon. The possibility exists for void formation and growth, depending upon the relative rates of snowpack settlement and slope-parallel creep, coupled with the necessity of a structurally weakened basal plane zone. Also, evidence may exist on the flank region fracture profile that may correlate with dominant mode buckling pending further study and data acquisition
Since buckling is a structural phenomenon extending into the toe region of the slope, physical measurement of void formation and buckle mode growth is difficult. Evidence of buckling from surface measurements is probably inconclusive because of wind transport of material that would obliterate long term geometry changes. Thus, advanced in-situ techniques such as acoustic emission monitoring, or slope modeling techniques, may be required to further assess the toe region influence on avalanche release.
*Department of Civil Engineering and Engineering Mechanics, Montana State University, Bozeman, Montana.
1. Sommerfeld, R. A., The role of stress concentration in slab avalanche release, Journal of Glaciology, 8, 451, 1969.
2. Bradley, C. C. and D. Bowles, The role of stress concentration in slab avalanche release: Comments on Dr. R. A. Sommerfeld's paper, Journal of Glaciology, 9, 411, 1970.
3. Haefeli, R., Stress transformation, tensile strengths and rupture processes of the snow cover, Ice and Snow, p 141, MIT Press, Cambridge, Mass., 1966.
4. Roch, A., Les desclenchements d'avalanches, Proceedings International Symposium on Scientific Aspects of Snow and Ice Avalanches, Gentltugge, Belgium, Intern. Ass. Sci. Hydrol., p 182, 1966.
5. Perla, R. I., The slab avalanche, 99 pp., U.S. Department of Agriculture, Alta Avalanche Study Center, Alta, Utah, Report No. 100, 1971.
6. Brown, C. B., R. J. Evans, and E. R. LaChapelle, Slab avalanching and the state of stress in fallen snow, Journal Geophysical Research, 77, 4570, 1972.
7. Lang, T. E., R. L. Brown, W. F. St. Lawrence, and C. C. Bradley, Buckling characteristics of a sloping slab, Journal Geophysical Sciences, in press, 1972.
8. Shinojima, K., Study on the viscoelastic deformation of deposited snow, Physics of Snow and Ice, Proceedings of International Conference on Low Temperature Science, Sapporo, Japan, p 875, 1967.
9. Yosida, Z., Physical properties of snow, Ice and Snow, p 485, MIT Press, Cambridge, Mass., 1966.
10. Brown, R. L., T. E. Lang, W. F. St. Lawrence, and C. C. Bradley, A failure theory for snow, Presented at 3rd international discussion conference on the general principals of rheology, Prague, Czechoslovakia, September, 1972.
The authors wish to express thanks to Dr. Glen L. Martin, Head, Department of Civil Engineering and Engineering Mechanics, Montana State University, for his interest in and support of the work reported.
The participation of Dr. Charles C. Bradley and Mr. William F. St. Lawrence in support of this work is also gratefully acknowledged by the authors.