Introduction

In the progression of analyses of steeply sloping snow-packs, emphasis has been placed upon the state of stress in attempts to understand the mechanisms leading to avalanche release. Reference Haefeli and KingeryHaefeli (1963) hypothesizes a material transformation based upon change of the axes of principal stress with depth leading to deep slab failures. Reference PerlaPerla (1971) develops a proof of the existence of a weak basal plane as a prerequisite for avalanche release, drawing upon the observed characteristics of the crown-region fracture geometry, and the associated stress distribution. Reference Smith and CurtisSmith and Curtis ([1975]) demonstrate the applicability of numerical methods to slab analysis, and model stress distribution and failure prediction based upon material layering and local geometric effects. A first attempt at deformation prediction was made by Reference Lang and BrownLang and Brown ([1975]) in which a nominal visco-elastic material idealization is used in conjunction with an assumed shear weakness in the basal plane to predict stress and strain-rate distributions theoretically. The present work is the next logical step in continued analysis of steep snow-packs. Analytically predicted snow-pack deformations are compared to actual measured deformations obtained by use of strain gages mounted in creeping snow-packs.

The ability to represent the deformation state of snow-pack analytically, a more difficult task than stress prediction, opens numerous possibilities for greater understanding of the avalanche problem. For example, correlation with developing theories of the incidence and geometry of failure planes, strain-rates for fracture, influence of weak layers on creep, and other phenomena of snow creep becomes feasible.

Snow-packs in avalanche paths have irregular surfaces of contact with the ground, are layered, and exhibit large variation in specific weight with attendant large variations in material properties. Snow is also orthotropic in the sense of exhibiting different material properties in tension and compression. Complexities of this sort are readily handled by the finite-element method, which makes this technique a logical choice for snow-pack modeling.

In this paper the authors report the following: (1) the construction of a finite-element analytical model; (2) a careful estimation of material properties from available data; (3) the first results from an extensive deformation-measuring program; and (4) a detailed comparison between the modeled and measured deformations.

The instrumentation used in the field studies has been described by Reference SommerfeldSommerfeld ([1975]). The deformation measurements were made during the course of three winters at Berthoud Pass, Colorado. In this paper we use the results of the first 25 d of measurements in which the positions of four points in the snow-pack were monitored. The analysis can be viewed as a pilot study to determine the feasibility of modeling deformation of sloping snow-packs.

The Finite-Element Model

A finite-element model is constructed which incorporates the plane-strain idealization, proportionality between stress and strain-rate (steady state) and a representation of material orthotropy. An efficient computer code that contains these features is completely documented by Reference Goudreau, Goudreau, Nickell and Dunham.Goudreau and others (1967). It is adapted for use on the CDC 7400 computer in order to carry out the calculations presented here. The computer code accepts a maximum of 25 different material descriptions, which is the principal limitation in application to the slope deformation problem.

The model geometry is derived from the experimental geometry but is not an exact duplication. The layers of the model do not necessarily correspond to the measured snow layering, primarily because of the material limitation noted above. Therefore, specific weight and secondary creep-rate coefficients are linearly weighted in the proportion of each experimental layer thickness in each finite element (compare Figure 1 with Figure 2).

Fig. 1. Detail of slope geometry and finite-element representation.

Fig. 2. Specific weight of snow versus height at the four gaging stations, with superimposed levels of finite-element layering as indicated. Data for day 347, 1972.

Three different periods are modeled. A pre-storm period, days 350 to 352, 1972, is modeled with a five-layer model (Fig. 1); a storm period, days 353 to 357, is modeled with a seven-layer model; and a post-storm period, days 001 to 005, 1973, is modeled also with the seven-layer model.

For stress evaluations, local geometry variations are not of extreme importance. However, it was found that the detailed geometry of the snow-ground surface has a pronounced effect on the deformations. A surface with four constant-slope segments proved inadequate. The gradually changing slope geometry which was adopted as shown in Figure 1, significantly reduces numerical perturbations. Gradual changes in material properties along the layers proved also to be necessary to eliminate large deformation perturbations between successive elements. This requirement for gradual changes in specific weight resulted in the use of many more material specifications than originally anticipated.

Snow has different material properties in tension and compression Reference Shinojima and Ōura(Shinojima, 1967) which can be represented by assuming local material orthotropy. However, prior to evaluation, the distribution of tension and compression components is not known, so an iterative procedure is used to account for this orthotropy. The stress in all finite elements is computed using a totally compressive material description. The material description is then modified based upon the tensile components obtained. The stresses are recomputed based upon this up-date, and the cycling continued until sufficient agreement is achieved. After three or four iterations of this type, the local angles of orthotropy, which are used as an index of convergence, tend toward constant values.

Output from the computer code includes components of steady-state deformation rates at each grid point, stress components and corresponding strain-rate components for each element, the local angle of orthotropy, and the internal energy rate.

Material Representation

The material representation for snow should account for both volumetric and deviatoric components of deformation. For each of these mechanisms of deformation a four-element Burgers-body representation is assumed in which stress is related to strain by

(1)

In this general form of the equation σ and ε may be either the dilatational or deviatoric components of stress and strain respectively. Proceeding with the standard process of taking the Laplace transform and retaining only the steady-state response to a step input of stress, the components of strain-rate are found to be related to the stresses, in a plane-strain representation, by

(2)

Details of the derivation leading to Equation (2) are outlined in a paper in preparationFootnote *. In Equation (2)

(3)

where K and μ are the bulk modules and shear viscosity for snow respectively. Separate values for these constants must be established for snow in compression and tension. To establish values for K and μ, results by Reference Shinojima and ŌuraShinojima (1967) and by Reference Lang and BrownLang and Brown ([1975]) are used for the compression relationships, namely,

(4)

and

where γ is the specific weight of the snow and T is the Celsius snow temperature. The corresponding relationships for tension are derived from results by Reference Shinojima and ŌuraShinojima (1967) and by Reference SalmSalm ([1975]), namely,

(5)

and

where these equations are applicable for applied stresses less than 8000 N m^-2. It is noted from Equations (4) and (5), by virtue of the relative magnitude of the amplitude coefficients, that separate accounting of principal tensile and compressive axes in the material representation introduces a significant difference in material viscosity. By solving Equations (4) and (5) for K and μ and substituting these values into Equation (2), separate constitutive equations for snow in tension or compression can be established. To represent the case of combined tension and compression components at a point, we start with the three-dimensional orthotropic material law, which for tension along the x axis and compression along the y and z axes, is

(6)

Introducing the condition of plane strain (ε_z = 0) and noting the correspondence

(7)

the modified constitutive law relating strain-rate to stress, for steady-state response, is

(8)

where

(9)

The coefficients in Equations (9) are not independent, but are subject to the reciprocity relationship of linear orthotropy that v_TE_C=V_CE_T. By restricting attention to steady-state response, Equation (8) contains constant coefficients only, so that a computer code set up for elastic equations can equally well be applied to secondary-creep calculations by using the coefficients of Equation (8) rather than elastic coefficients.

Experimental Program

Three types of gage were designed to be placed in the snow-pack: a vertical gage to measure settlement, a slope-parallel gage to measure differential creep, and an angle gage to measure creep angle. The size of the linear gages was based upon the statistical evidence of Reference SommerfeldSommerfeld ([1975]) that characteristic lengths of 1.0 m can be associated with nominal snow properties.

The vertical deformation gages were approximately 1.0 m high (Fig. 3). Each gage consisted of a sensing element, an actuation rod, an actuation head, and a protective housing. The sensing element is a linear potentiometer with 1000 Ω total resistance, a linearity of 0.1%, and a total range of 150 mm. The actuation rod (12 mm diameter, aluminum) is attached to the potentiometer at the lower end and to the actuation head (12 mm wire mesh, 0.1 m x 1.0 m rectangle) at the upper end through a swivel joint. The swivel allows rotation between rod and actuation head in the vertical plane, this angle measured by a rotary potentiometer with a total resistance of 1 000 Ω, a linearity of 0.1% and a range of 90°.

Fig. 3. Deformation-gage notation and geometry in the snow-pack as set on day 341, 1972.

The 3.0 m long slope-parallel gages contain the same type of sensing element as the vertical gages. For the slope-parallel gages an actuation head of wire mesh, as above, was attached at each end, and the connecting actuation rods (made from steel to match thermal expansion with that of polycrystalline snow) rode in “Teflon" bearings at each end. The connecting rods were given a slight upward bow so that the rods did not contact the protective housing of the sensors.

Each sensing element formed one half of a bridge circuit, the other half being housed in a heated instrument building about 200 m away and connected by shielded instrument cable. The bridges were powered by regulated 20 V d.c.supplies. Sensor outputs were recorded on a slow-speed magnetic tape recorder with a range of +1.0 V. The bridges were periodically balanced to keep them within the recorder range.

Four vertical gages were installed before the first snow. They were bolted to steel spikes which had been driven into the ground and oriented so that downhill creep loosened the bolts. The plan was for snow to fill in around the gages, but a small avalanche early in the first large storm knocked down the three lowest gages. In resetting these the snow was severely disturbed.

On day 341, 1972 the gages were installed in approximately 1.0 m snow. Exact heights and gage designations are shown in Figure 3. The gage outputs were monitored continuously from that time. The slope-parallel gages operated throughout the winter, but the vertical gages reached the ends of their travels at different times. When this happened, the creep (rotation) gages were also adversely affected.

The analog data were digitized and corrected for calibration constants and bridge reset displacements. The final form was in millimeters from initial separation or in degrees from initial angle, listed every two hours. These data are plotted in Figure 4. initial erratic movement of the gages is attributed to snow resettlement after gage installation.

Fig. 4. Linear and angle transducer response curves for the ten gages of the bridge array from day 341, 1972, to day 005, 1973.

The slope geometry was determined by a transit survey using a meter stick as a rod. The confirmed accuracy of the survey is ±50 mm. The snow-pack geometry was monitored by means of snow cores taken with a Federal snow sampler. The cores were supplemented by snow pits. A compromise was necessary between obtaining accurate data on the geometry and causing excessive disturbance to the snow. Also, the type of corer used precluded accurate density measurements of the temperature-gradient snow that occurred at the bottom of the pack.

Analytical and Experimental Results

The first time period considered for analysis was day 350 through 352, 1972. By this time, the erratic behavior associated with installation had subsided, no new snow had fallen, and the pack could be considered in a steady state. A set of cores was taken on day 347 and these data were used to establish the parameters of the five-layer model (Fig. 2). Since the samples of the temperature-gradient snow were likely to introduce errors, the densities of the bottom layer were established using Reference Judson and ŌuraJudson’s (1967) climatological data (shaded regions in Fig. 2). It was found that the bottom layers were important in determining the flow of the pack. For this reason, and because the specific weights of the bottom layer were not well known, two additional computer runs were made with the density of the lowest layer varied by ± 10%.

The linear deformation rates in mm/d and the angular deformation rates in deg/d were taken from the two-hourly deformation tabulations. Comparisons between the computed and the measured deformation rates are listed in Table I. Rates of gages Δɸ₁, and Δl₁ were inoperable at this time. On days 353 to 357, 0.53 m of snow with density 1 600 to 1 800 N m^-3 deposited in the experimental area. This period was modeled using the stratigraphy of day 347 with the superposition of two additional layers to represent the new deposition. A comparison of the measured and calculated rates is shown in Table I.

TABLE I. COMPARISON OF EXPERIMENTAL MEASURED AND ANALYTICALLY COMPUTED DEFORMATION RATES AT VARIOUS DAYS OVER THE DURATION OF THE EXPERIMENT

The snow stratigraphy measured on day 005 (Fig. 5) is used to formulate another seven-layer model for days 001 to 005, 1973. By day 001, over half the gages had reached the limits of their ranges. Extrapolated deformation rates based on the slopes of the curves in Figure 4 were used in these cases.

Fig. 5. Specific weight of snow versus height at the four gaging stations, with superimposed levels of finite-element layering as indicated. Data for day 005, 1973.

It became evident that the specific description of the basal layer has a marked effect on the creep response of the entire slope. It should be noted that the basal layer, which varied in vertical depth between 50 and 100 mm, was well developed depth hoar (III-B-2 and III-B-3) during the experimental period. The mechanical properties of this type of snow are virtually unknown. For this reason two computer runs were made with the model of Figure 1, but with basal-layer density changed by +10% and − 10%. The deformation values obtained are compared with the initial model inTable II.

TABLE II. VARIATION IN THE DEFORMATION RATES FOR A ± 10% VARIATION IN SPECIFIC WEIGHT SPECIFICATION OF THE BASAL LAYER (100 mm) OF THE FIVE- LAYER ANALYTICAL MODEL

Summary and Conclusions

Comparison of the experimental and analytical deformation rates listed in Tables I and II demonstrates a close order-of-magnitude correspondence. The analytical deformation rates for days 350 through 353 and 353 through 357, 1972, are larger than the experimental values, showing a more fluid analytical representation than that exhibited by the slope. Much of the discrepancy can be attributed to the problems of defining the basal layer accurately. The strong influence of the basal layer on creep and settlement is demonstrated by the results of the sensitivity analysis shown in Table II. A ±10% variation in the specific weight of the basal layer results in a ±28% variation in creep and settlement rates 1.0 m above the ground in a snow slab averaging 1.2 m depth. In the analytical model, the viscous properties of normally metamorphosed mid-pack snow of comparable specific weight were used. It is generally believed that higher-density depth hoar, is stiffer than comparable mid-pack snow before failure. This is consistent with the fact that the deformation rates corresponding to the + 10% case more nearly coincide with the experimental values than do the rates deduced from the nominal densities established from the climatological data.

The deformation rates computed for days 001 through 005, 1973, are based on a second stratification measurement carried out on day 005. The analytical model verifies the observed trend that the slab was stabilizing because of the densification of lower layers. It is likely that this densification process was proceeding continually throughout the experiment, which would account for the greater total discrepancy between analysis and experiment on days 353 through 357, compared with days 350 to 352. For both of these periods, the material description was based on the stratigraphy determined on day 347 without introducing any factor to account for densification. A densification process is not in strong evidence in the deformation histories of Figure 4, so the effect cannot be considered dominant. The apparent densification suggested by the history of gage ɸ₄ (Fig. 4) is an artefact caused by the bottoming of gage l₇, during day 346. After this date gage ɸ₄ changed more slowly since it also reacted to differential settlement. Similar behavior pertains to other gages on days 001 through 005, as more than half the gages had reached the end of their travel by day 001.

Gross adjustment of the specific weight of the entire basal layer by some factor does not eliminate all of the discrepancy between experimental and analytical results. Of particular interest is the lack of correlation between the slope-parallel gages, denoted by rates Δl₂,Δl₄ and Δl₆. The creep rates reflect changes in lengths of the material fibers parallel to the slope. For an infinite uniform slope these rates are, in fact, zero, so that material inhomogenity and irregular geometry must contribute to the non-zero values. Due to the extreme sensitivity of the creep to the character of the basal layers, outcrops with dimensions in excess of the basal layer thickness (≈ 100 mm) could have a significant effect on local flow rates, particularly on the slope-parallel strain-rate. In the analytical modeling a smooth ground surface was assumed.

A second possible cause of the discrepancies is that the stratification measurements made at the four gage stations are not close enough to detect all of the local lens depositions that are numerous in the test field due to wind-induced deposition. Whereas, for stress prediction, detailed terrain specification and material stratification are secondary effects, the same is not the case if one wishes to obtain a detailed model of the deformation states. It was observed that in smoothing the distribution of specific weight in a particular layer, significant changes in deformation rates occurred also. Thus it is to be expected that local lenses of high or low density will influence local deformation rates.

This discussion by no means exhausts plausible explanations for local perturbations between experiment and analysis. However, the overriding point from these results is that general correspondence is established between experiment and analysis of a typical alpine slope. The dominant slope-deformation characteristic is that flow is sensitive to the presence of a well-defined weak layer, and accurate modeling is dependent on explicit definition of the flow properties of this type of layer. A second important conclusion is that the credibility of supportive analyses of slope geometries is established, if due regard is given to the determination of material stratigraphy. Based, in part, on these findings, the suggestion is made that correlating slope analyses should be considered as a viable tool in establishing and studying characteristics of snow-avalanche mechanics.