2.1. Performance of the tests

Reference SigristSigrist (2006), Reference GauthierGauthier (2007), Reference Gauthier and JamiesonGauthier and Jamieson (2008) and Reference McClungMcClung (2009b) provide detailed descriptions of the field tests used in this paper. Of the field test data, >90% are from the work of Reference GauthierGauthier (2007) and Reference Gauthier and JamiesonGauthier and Jamieson (2008). The procedure involves introducing a cut upslope or downslope within a weak layer underneath a long rectangular block of snow cut free on all sides. The cut is made to a critical length, l, at which the shear fracture propagates rapidly within the weak layer. Figure 1 contains a schematic of the test set-up. The block of snow is 30 cm wide with a total length L ₀ = 1 m and ideally more than two times D. In most of the tests, all sides are free surfaces.

Fig. 1. (a) Schematic for field tests made with a saw cut of critical length L within a weak layer of thickness d ₀ with a total block length L ₀. (b) Side-view schematic of the field tests, with D as the slab depth and slope angle ψ. The parameter ω is the length of the fracture process zone. The coordinate system is defined such that x is measured from the left end of the block (x = 0) to the end of the cut or slip surface (x = L) in the direction of the dotted line in the centre of the weak layer.

The cut within the weak layer (Fig. 1) is made either upslope from the lower end of the block or downslope from the upper end of the block. Reference Gauthier and JamiesonGauthier and Jamieson (2008) showed that there is no statistical difference between critical cut lengths made upslope or downslope. For the remainder of this paper and in the model presented later, I take the cut as being made upslope. With the cut starting from the free surface at the downslope end of the block, it is important that when the propagation condition is met, the cut is not close to the free surface at the upper end of the block. The standard length of the block is 1 m and median cut lengths do not exceed 0.65 m, so this condition is normally fulfilled.

One important feature of the tests (Fig. 1) is that the weak layer must be thick enough so that the saw cut does not intersect the slab. Thus, the weak-layer thickness, d ₀, should be greater than ∼5 mm. The finite thickness of the weak layers tested, along with the finite thickness of the saw, gives rise to a two dimensional (2-D) deformation pattern in the weak layer with slow, slope-normal deformation working to bring the crack faces together. The 2-D aspect negates comparison of the test results with the earlier models (e.g. Reference McClungMcClung, 1981) based strictly on slope-parallel deformation only in the weak layer. Avalanches sometimes release by shear fracture on weak layers which are on the order of 1 mm thick. Examples include layers of 2-D stellar crystals and surface hoar ∼1 mm thick. The analysis in this paper does not apply to these cases.

Table 1 contains descriptive statistics for field-measured parameters from 42 slab/weak-layer combinations compiled from 559 tests where ρ is the mean slab density. All the tests were done on dry snow.

2.2. Observations from high-speed films of the propagation saw tests

When a saw cut is made within the weak layer, there are some important features which may not apply to avalanche release but are important in interpreting the test results. Highspeed films (300 frames s⁻¹) and precision displacement, speed and acceleration estimates were made using particle tracking (0.11 mm pixel⁻¹) during the saw cuts. The particle tracking was done with particles attached to the slab. Maximum slope-normal displacement prior to propagation was <1 mm. Slope-parallel displacement in the slab was not observed. This is probably due to the resolution accuracy. In addition, the saw cut will produce the overwhelming amount of displacement in the weak layer, not the slab, and weak-layer displacements were not measured. Model predictions (section 7) suggest the total weak-layer slip, δ ₀, is on the order of a few tenths of a millimetre.

Two types of initial fractures are observed with the tests. In most cases, slope-parallel weak-layer propagation is observed. However, slab tensile fracture is sometimes observed, occurring at a distance of ∼1 D ahead of the saw. Reference GauthierGauthier (2007) presented extensive data from 87 slab/weak-layer combinations representing >800 tests with a 2 mm thick saw. The data showed that in 16% of the combinations, tensile slab fracture was primarily observed in the individual tests; in 71%, weak-layer propagation was primarily observed; and in 13%, observations were missing or inconclusive or the mix of slab fractures and slope-parallel propagation was about the same. Thus, in most cases, the first fracture to propagate is in the slope-parallel direction which is interpreted here as mode II fracture.

The films and observations revealed that test results depend on the blade thickness of the saw used. With a thin saw (1 mm), the first fracture to propagate is nearly always in the slope-parallel direction. With a thicker saw (3 mm), the first fracture still usually propagates in the slope-parallel direction but tensile slab fractures occur more often than with the thin saw. For cases of tensile slab fracture, I believe that the gap created by the saw causes slab bending to precipitate the slab tensile fracture.

Tensile slab fractures in slab avalanches (Reference McClungMcClung, 2009a) occur, on average, at a distance of ∼50D after initiation, which is long after weak-layer propagation. Thus, the initial slab fractures seen in some of the propagation saw tests (at 1 D) seem to be an artifact of a gap created by the saw rather than having much relation to avalanche tensile fractures.

In all cases, the films showed that the collapse mentioned in the models of Reference Perla and LaChapellePerla and LaChapelle (1970) and Reference Heierli, Gumbsch and ZaiserHeierli and others (2008) is a dynamic effect that occurs after, and in response to, the shock of either dynamic slope-parallel weak-layer propagation or dynamic tensile slab failure. Since the gap created by the saw is unlikely to be realistic for avalanche release, slope-parallel propagation, as seen in most of the tests, is assumed to be the initiation mechanism for fracture.

Results from high-speed films of the tests show there is very little contact of the faces of the crack behind the saw cut. Reference HaefeliHaefeli (1951, p.195; 1954, p.101–102) claimed to have measured the residual shear stress for alpine snow and τ _r ≈ 0. Given these results, it is assumed that τ _r ≈ 0 for the analysis in this paper. This assumption might not apply to avalanche initiation for which there is no saw cut.

4.1. Determination of for snow in simple shear laboratory experiments

Laboratory test results show that when failure is achieved during slow load application (a peak on a shear-stress/displacement curve) alpine snow is in a dilatant state whether the sample simply fails and slowly softens after the peak, or fractures and collapses or settles rapidly after the peak is achieved (Reference McClungMcClung, 2007c, Reference McClung2009b. The laboratory results suggest that the characteristic mean displacement over which rapid softening takes place in a fracture experiment is of the order . This value is similar to results in tension fracture experiments on alpine snow (Reference McClung and SchaererMcClung and Schaerer, 2006, p.84; Reference SigristSigrist, 2006; Reference Borstad and McClungBorstad and McClung, 2009), and concrete (Reference Bažant and PlanasBažant and Planas, 1998, p.180). Slow shearing strain-softening failures on alpine snow without fracture typically show displacements over which softening takes place an order of magnitude above this value, i.e. or more (Reference McClungMcClung, 1977; Reference SchweizerSchweizer, 1998; Reference McClung and SchweizerMcClung and Schweizer, 1999).

The smaller displacement in shear during fracture, as opposed to slow softening, is of immense importance since the size of the FPZ is determined mostly by the material parameter relatively independent of the size of the snow slab (Reference Palmer and RicePalmer and Rice, 1973). In the present study, the assumed value is , which is an appropriate value for shear fracture experiments on dry snow during shear fracturing (Reference McClungMcClung, 2009b). Precision estimates of with closed loop testing in shear are not yet available for alpine snow.

4.2. Determination of values of μ ^* for weak-layer forms from field tests

In the present model, the effective values of μ ^* represent the pressure sensitivity for weak-layer forms by the proportion of slope-parallel traction accompanied by rapid slope-parallel weak-layer displacement as the saw cut is made. The traction is assumed to be the initial stress released by the saw cut. It is taken to be a fraction of the slope-normal stress at the weak layer: τ ₀ = μ ^* σ _n (Reference McClungMcClung, 2009b), which must be regarded as an approximation.

When a saw cut is made rapidly, as in the present experiments, fracture is rapid without the dilatant grain rearrangement during the slow laboratory shear tests. The values of μ ^* applied in the model verification below were taken from pressure sensitivity tests on avalanche weak layers reported by Reference JamiesonJamieson (1995), who placed weights on shear frames. The values were calculated by making linear fits through the shear frame strengths measured as a function of applied normal stress. These calculations gave the following values for μ ^* for different crystal forms: 0.20 (surface hoar), 0.37 (facets), 0.07 (rounded facets) and 0.47 (decomposing and fragmented forms).

Independent confirmation of these values is provided by Reference Van Herwijnen and HeierliVan Herwijnen and Heierli (2009). They reported similar instantaneous, initial values just after failure but before frictional contact, which follows dynamic layer collapse. From experiments on surface hoar, faceted crystals and depth hoar from propagation saw tests, skier-tested events and rutschblock tests, reported values mostly range from about 0.2 to 0.4, with one very low value of ∼0.02. All these values are far below pressure sensitivity estimates from slow tests (Reference McClungMcClung, 1987) which do not exhibit fast fracture.

Reference HaefeliHaefeli (1954, p.116–119) reported similar experimental data on depth hoar. The tests were performed rapidly with duration of a few seconds since the rate of load application was ∼0.6 kPa s⁻¹ and maximum shear strength was ∼2 kPa. A least-squares fit to the mean values of strength (mean value for five series of tests) with normal loads from 0 to 3 kPa give a value μ ^* = 0.46. Further refinement, taking into account changes in cohesion with normal stress according to Haefeli, gave μ ^* in the range 0.27–0.36. These results are in good agreement with the tests of Reference JamiesonJamieson (1995) for the other persistent forms (surface hoar and facets) and the results of Reference Van Herwijnen and HeierliVan Herwijnen and Heierli (2009).

4.3. Weak-layer mode II fracture energy, G _II

In the present study, the weak-layer fracture energy is taken as a constant, median value estimated by Reference McClungMcClung (2007c) for avalanche applications. The values (Reference McClungMcClung, 2007c) were determined from slab avalanche fracture line data using the mode II stress intensity factor determined by Reference Bažant, Zi and McClungBažant and others (2003) assuming elastic slab behaviour. Due to the rate dependence of alpine snow, the present tests are too slow to be assumed elastic, so it would not be appropriate to determine G _II along with the tests. The range of G _II calculated from 278 slab avalanche profiles (Reference McClungMcClung, 2007c) is 0.001–0.2 J m⁻². The median value determined by Reference McClungMcClung (2007a), G _II = 0.02 J m ⁻², is used in the present validation. This is comparable to, but lower than, values determined by Reference SigristSigrist (2006) (0.07 J m ⁻²) and Reference Sigrist and SchweizerSigrist and Schweizer (2007) and assumed by Reference Heierli, Gumbsch and ZaiserHeierli and others (2008) (0.03–0.07 J m⁻²).

4.4. Viscoelastic and rate-dependent modulus, E’

Alpine snow is a highly rate-dependent material (Reference MellorMellor, 1975; Chae, 1967; Shinojima, 1967) and it is required that deformation must be very rapid in order to avoid viscous effects. The experiments described in this paper have been performed with the speed of the saw cut on the order of 10–20 c ms⁻¹ (Reference GauthierGauthier, 2007; Reference McClungMcClung, 2009b), with time to fracture of several seconds. According to Reference Palmer and RicePalmer and Rice (1973), Rice (Reference Palmer and Rice1973, p.272) and Reference BažantBažant (2005, p.156), the slab modulus for viscoelastic deformation should be chosen in relation to the frequency implied by the speed of advance over the scale of the FPZ.

From the work of Reference SigristSigrist (2006) and Reference Borstad and McClungBorstad and McClung (2009), it may be inferred that the scale of the FPZ in rapid tension fractures is about 5–10 cm, which is similar to the saw width in the present work. Such estimates are for slab material with, normally, smaller grain sizes than for weak-layer forms. Thus, it is possible that, on average, the FPZ for the weak layers used in this paper (mostly surface hoar) may be larger than 5–10 cm. For application of the saw tests to shear fractures, the speed of advance relative to the FPZ size implies that the relevant frequency is of order 1 Hz for choice of the modulus.

For this paper, the viscoelastic storage shear modulus of the slab, μ _s, at a frequency of 1 Hz as a function of mean slab density, ρ (kg m⁻²), is determined by a nonlinear fit through the data of Reference Camponovo and SchweizerCamponovo and Schweizer (2001) from Reference McClungMcClung (2007a) as:

The field data are all taken from weak layers with finite thickness d ₀, on average, ∼2 cm. For a layered system, it is shown by Reference Hutchinson and SuoHutchinson and Suo (1992) that the effective shear modulus, μ, for describing fracture varies between μ _wl < μ < μ _s, where μ _wl is the effective modulus for the weak layer. The upper limit, μ _s, applies for an infinitesimally thin crack adjacent to the slab, and the lower limit, μ _wl, applies for a thin crack entirely embedded within a thick weak layer. For the present experiments, the weak layer is thin, but of finite thickness, and the saw cut is not infinitesimally thin. so the effective modulus should be between the limits. From hardness differences between the slab and weak layer, it is suggested that the weak layer is typically lower by a factor of 10 or more in modulus than the slab in avalanche work. Thus, it is possible that the appropriate modulus, μ, to describe the fracture may vary from the slab stiffness to a factor of ten or so lower than the slab stiffness.

For the present paper, the shear modulus is assumed to vary as with a constant stress model from Reference Hull and ClyneHull and Clyne (1996) for a composite material. For a slab with a weak layer occupying a fraction, f, of the slab/weak-layer system, mixture theory gives the effective shear modulus as

where f = d ₀ = D, μ _wl is the effective modulus of the weak layer and d ₀ = 0.02 m is the average weak-layer thickness. It is assumed that the weak layer has a modulus of μ _wl = (1/20) μ _s in the model calculations below. This assumption produces an effective modulus appropriate for the experiments of, on average, about half μ _s appropriate for the slab of given average density. For a median value of D, f = d ₀/D = 0.02 m/0.33 m = 0.06 and slab modulus 20 times the weak-layer modulus, Equation (6) yields μ = μ _s/2:14.

The constant stress mixture model contained in Equations (5) and ( 6) yields the limits predicted by Reference Hutchinson and SuoHutchinson and Suo (1992) for the special cases they considered. If d ₀/D → 0 then μ → μ _s, for the case in which an infini tesimally thin crack is adjacent to the slab. If d ₀/D → 1, then μ → μ _wl for the case in which an infinitesimally thin crack is entirely within a weak layer. The high-speed films (300 frames s⁻¹) of the tests and the work of Reference SigristSigrist (2006, p.90) have shown that the slope-normal deformation within the saw cut is not infinitesimal as assumed in the linear elastic fracture mechanics (LEFM) model of Reference Hutchinson and SuoHutchinson and Suo (1992). Nevertheless, their work suggests reasonable bounds on the effective modulus. Of the two bounds, μ → μ _s is the most unrealistic, since the crack is neither infinitesimally thin, nor directly adjacent to the slab. In order to perform the tests, so that the saw cut does not enter into the slab, the layer thickness must be greater than about d ₀ = 5 mm such that the saw cut remains entirely within the weak layer.

The effective modulus used in this paper for mode II fracture is

where v is the analog Poisson’s ratio for viscoelastic correspondence. For the low-density snow in the experiments (85–266 kg m⁻³), based on results suggested by Reference MellorMellor (1975) and summarized by Reference SalmSalm (1977), it is assumed that v = 0.1. This corresponds to the upper limit of the data of Shinojima (1967).

Equation (7) represents an important assumption in the analysis below. It is partially justified by the model success explained below and the fact that the effective modulus for the layered system (slab/weak-layer) should be less than that for the slab for weak layers of finite thickness, as with the experiments. In the experiments, the saw is forced through the weak layer and energy transfer takes place both in the slab and weak layer.

5.1. Direct model comparison with data

Table 2 contains a summary of input parameters used in the model comparison for the four input parameters not measured in the tests. The parameter E = 2μ(1 + v) = 2.2μ is the implied rate-dependent Young’s modulus, which may easily be compared with published values (Reference Schulson and DuvalSchulson and Duval, 2009).

Figure 3 shows comparison of the data for 27 weak-layer/slab combinations (355 tests) for which the weak-layer crystal forms are known. Nonlinear regression modelling showed

where C = 1.01 ± 0.13 within 95% confidence limits and percentage variance explained as R ² (Predicted/Observed) = 0.76. There is considerable scatter in Figure 3. This is partly due to natural variations in the data with two input parameters ( , G _II) left constant.

Fig. 3. Scatter plot of calculated versus measured values of L/D for 27 slab/weak-layer combinations representing 355 tests. The constant of proportionality is within 95% confidence limits.

5.2. Implied probability density function

Figure 4 shows the experimental data for L/D fitted to a Gumbel PDF. The data represent 45 different weak-layer/slab combinations and 563 tests. The abscissa is the Reduced Variate defined as –ln (–ln (P _N) , where P _N is the non-excedance probability given by and f_X (x)(x = L/D) is the PDF. A Kolmogorov–Smirnov (KS) goodness-of-fit test gave the K-S test statistic as 0.103 with a p value of 0.721. The location parameter and scale parameters are 0.76 and 0.39 respectively.

Fig. 4. Probability plot for experimental values of medians for 45 slab/weak-layer combinations, weak layers representing 563 tests. The plot suggests that the ratio follows a Gumbel normal PDF. The reduced variate is –ln(–ln(P _N)), where P _N is the non-exceedance probability defined in the text.

Figure 5 shows calculated model values for 27 weaklayer–slab combinations (355 tests) fitted to a Gumbel PDF using the parameters above. The K-S statistic is 0.155 with a p value of 0.533. The location parameter is 0.70 and the scale parameter is 0.56.

Fig. 5. Probability plot for 27 slab/weak-layer combinations representing calculated vales for 355 tests similar to Figure 3. The plot suggests the values follow a Gumbel probability function.

Other distributions were considered including lognormal, Weibull, Frêchet and gamma, but the Gumbel PDF was chosen since it gave the best goodness-of-fit statistics and the best fit for the probability plot for both the sets of measured and calculated values. Thus, the choice of the Gumbel PDF is empirically based.

Due to lack of information about slab density, weak-layer crystal form or μ ^*, Figure 5 does not have as many weaklayer/slab combinations as Figure 4. The constant input parameters, (G _II = 0.02 J m⁻²; ) imply μ ^* σ _N<250 Pa in order to obtain a solution. Thus, some slab/weak-layer combinations did not yield solutions for high normal stresses (>1 kPa). The highest slope-normal stresses in the database come from faceted weak layers with slope-normal stress around 2500 Pa, which either implies μ ^*<0.1 or G _II >0.02 J m⁻².

Both of these explanations are possible. Reference JamiesonJamieson (1995) measured a weak layer with rounded, faceted forms which implies μ ^* = 0.07 so the model can still apply. Importantly, high slope-normal stresses are usually associated with weak layers of older snow which may have higher fracture energy values than the average used in the model comparison here. It is well known that the shear strength of weak layers tends to increase with normal stress overburden and time, provided they are not subject to high temperature gradients (Reference Chalmers and JamiesonChalmers and Jamieson, 2003; Reference Zeidler and JamiesonZeidler and Jamieson, 2006). Since shear strength is an important component of G _II (Reference Bažant, Zi and McClungBažant and others, 2003), it is expected that G _II increases as layers age and σ _N increases by snow loading. In deep alpine snowpacks, strong temperature gradients are usually found in the upper portion, not at deep weak layers.

Use of G _II = 0.1 J m⁻² yields predictions for any case in the database. Assuming G _II = 0.1 J m⁻² for two deep slabs (highest values of σ _N in the database) on a faceted layer (Reference Gauthier and JamiesonGauthier and Jamieson, 2008) gave L/D (measured; predicted) as: (0.29; 0.25) and (0.36; 0.55) for values of (D, ρ, ψ) respectively of: (1.3 m; 233 kg m⁻³; 23°); (0.98 m; 262 kg m⁻³; 0°).

5.3. General slope-normal dependence

Slope-normal dependence in the field data and the model is a direct consequence of the release of gravitational potential energy by making the saw cut in a relatively thick weak layer. In order to isolate one parameter in field tests, slope-normal dependence in this case, the slab and weak layer properties must be roughly the same for each slope angle.

Three datasets are available for testing slope-normal dependence: (1) tests on slopes of 0°, 30° and 38° reported by Reference Gauthier and JamiesonGauthier and Jamieson (2008) for a weak layer of decomposing and fragmented crystals (DF); (2) tests on a surface hoar (SH) layer at Rogers Pass, British Columbia, Canada (Reference McClungMcClung, 2009b); and (3) tests on a surface hoar (SH) layer at Kootenay Pass, British Columbia (Reference McClungMcClung, 2009b). Table 2 contains a summary of test results in comparison with the model. The results show that the model predicts the correct trend and magnitude. For the same slab and weak layer, critical length increases with slope angle.

For constant slope angle (ψ = 30°), three sets of measurements with a weak layer of faceted snow reported by Reference SigristSigrist (2006) gave the following results for pairs of L/D (measured; predicted): (0.85; 0.67); (0.89; 0.73); (1.14; 1.10).

In order to explore the general slope-normal dependence further, Equation (4) may be expressed as

where and . Theoretical calculations of C ₁ for 35 slab/weak-layer calculations, with the inputs described above, show that it varies from 115 to 992 Pa, with a median value of 540 Pa. In order that the calculations may extend over the entire range of slope-normal stresses in the data (maximum value 2735 Pa; faceted snow), it is assumed that u ^* = 0.07 (minimum value for faceted weak layer) in C ₂ to give an estimate C₂ = 0.28 x 10⁻³ (Pa⁻¹) with , G _II = 0.02 J m⁻² as before. The normal stress calculations show that there is a lack of data between σ _N = 1–2 kPa, but the data suggest rapid decline in the value of L/D up until ∼1 kPa normal stress, and the model predicts similar behaviour.

Nonlinear robust regression of the data for Equation (9) (42 slab/weak-layer combinations), using the Marquardt algorithm with C ₂ = 0.28 x 10⁻³ Pa⁻¹, gave a value C ₁ = 293 Pa that falls approximately midway between the minimum (115 Pa) and median (540 Pa) values estimated theoretically. The percent variance explained for the nonlinear regression is (raw) R ² = 0.77. Figure 6 shows three calculated curves based on the theoretical minimum, maximum and median values of C ₁ compared with the measured data versus normal stress calculated from the data. The results show that the theoretical curves bracket the measurements well and the median curve goes approximately through the middle range of most of the data.

Fig. 6. Model limit estimates of L/D from Equation (7) with C ₂ = 0.28 kPa⁻¹ and C ₁ = 115 540 992 Pa as minimum, median and maximum

These numerical results are not sensitive to the chosen value of C ₂. If C ₂ = 0 then C ₁ = 286 Pa and if C ₂ = 0.36 × 10⁻³ Pa⁻¹ then C ₁ = 295 Pa. The latter value of C ₂ is the largest allowable for the dataset given that the highest value of σ _N is 2.375 kPa so that the constraint on Equation (4) is satisfied. The reason the calculations are not sensitive to C ₂ is the scarcity of data for σ _N between 1 and 2 kPa.