Solitary fracture waves in metastable snow stratifications
Joachim Heierli
WSL Swiss Federal Institute for Snow and Avalanche Research, Davos, Switzerland
Received 9 June 2004; revised 1 February 2005; accepted 28 February 2005; published 18 May 2005.
[1] Fracture processes in layered snow have many implications on avalanche hazard and mountain safety. This paper is a contribution for a better understanding of fracture dynamics in a stratified medium with a collapsible layer in terms of a simple analytical model involving only field measurable parameters. For this purpose a simple three-layer snow stratification is considered to consist of a compact basal layer, a collapsible weak layer, and a homogenous top layer. It is shown analytically that the fracture of the weak layer can occur in the form of a localized disturbance zone propagating a collapse with constant velocity and wavelength. Simple analytical expressions describing the disturbance are obtained. The results are in good agreement with observations in the terrain. The presented fracture process is relevant to the release of dry snow slab avalanches and to the propagation of firn quakes and whumpfs but may also apply to other stratified media with appropriate properties.
Citation: Heierli, J. (2005), Solitary fracture waves in metastable snow stratifications,J. Geophys. Res.,110, F02008, doi:10.1029/2004JF000178.
1. Introduction
[2] In mountainous regions, snow avalanches often represent a major natural hazard causing considerable human fatalities and losses. For many decades, avalanche forecasting and prevention has been an important factor in reducing the risk to population and infrastructure. Still, avalanche forecasting has remained a challenging issue.
[3] It is commonly accepted among snow scientists that dry snow slab avalanches are preceded by a fracture cutting through a buried weak layer or interface in the snowpack [Schweizer et al., 2003]. Such fracture processes are also observed in flat terrain. In perennial snow, these are frequently designated by firn quakes, whereas the term whumpf is often preferred in combination with seasonal snow stratifications.
[4] Several accounts of firn quakes and whumpfs describe sudden collapses with perceptible propagation velocity and direction. Firn quakes were observed to travel for many miles apparently unaffected by local anomalies in the snow stratification [Den Hartog, 1982]. Collapses of the snow- pack up to vertical 25 mm were reported [Sorge, 1933].
Whether a thin layer or a whole section collapsed was left unanswered. Sorge [1933] also hypothesized that firn quakes are the result of a sudden compression of the snowpack andTruman[1973] described this as a ‘‘domino effect’’ apparently occurring as ice particles assume a more tightly packed configuration. Few firn quake velocities were ever reported, but casual observations indicate a surprisingly wide range of propagation velocities between estimated 6 ms1[Truman, 1973] and the velocity of sound
in air, i.e., 310 to 330 ms1, depending on temperature [Den Hartog, 1982].
[5] Whumpfs and remotely triggered avalanches have also been associated with the compressive failure of weak snow layers consisting of buried surface hoar, depth hoar or facets [Johnson et al., 2000]. Indeed, fractured weak layers accompanied by a sudden settlement of the snowpack have been visually observed and reported on several occasions in combination with whumpfs [Johnson et al., 2005]. A photographically documented example of such a settlement, visible along a crack line, is shown by Jamieson and Schweizer [2000]. In one well-documented experiment at Bow Summit, Banff National Park, Alberta, the mean velocity of a propagating fracture was measured with geophysical equipment and an average speed of 20.0 ± 2.0 ms1 was reported [Johnson et al., 2000, 2005]. The weak layer collapsed by approximately 1 to 2 mm. This so far unique, invaluable experiment is an important test case for comparison with the model proposed in this work.
Johnson describes the phenomenon he observed with a homogenous wave equation and identifies the propagation velocity of the crack front with the velocity of flexural elastic waves in the slab [Johnson et al., 2000; Johnson, 2001;Johnson et al., 2005] and points out the essential role of gravitational potential energy in fracture propagation [Johnson, 2001].
[6] The primary purpose of the present paper is to provide a quantitative analysis of these phenomena by writing and solving the equation of collapse propagation in low-angle terrain. The solutions are localized, kink-shaped collapse regions propagating with constant velocity, reminiscent of solitary waves. The propagation velocity, its associated wavelength as well as the strain and strain rate fields are calculated and evaluated for Johnson’s Bow Summit experiment.
Copyright 2005 by the American Geophysical Union.
0148-0227/05/2004JF000178
[7] The paper is organized as follows. In section 2, two distinct types of disturbances are distinguished by the direction of the displacements with respect to the propa- gation plane (transverse and in-plane). In section 3, the partial differential equation for transverse crack propaga- tion is derived from linear elastic theory. Conditions are found under which stationary solutions are admissible. An analytical solution is derived and expressions for propa- gation velocity, wavelength, strain and strain rate are deduced. In section 4, experimental results from test cases are presented. In section 5, the observable predictions of the model are compared with the observations of the test cases. Section 6 consists of a summary and concluding remarks.
2. In-Plane and Transverse Disturbances
[8] A given snow stratification is metastable when a certain amount of gravitational potential energy is suddenly recoverable. For example, when a surface layer or slab is carried by a weak layer composed of hoar crystals for example, gravitational potential energy is suddenly gained when the weak layer collapses in a small area. Similarly, when a surface layer or slab is bonded by a weak interface to a tilted basal layer, gravitational potential energy is gained when the interface fails in a small area and the slab readjusts by a slide (without necessarily any rupture on the perimeter of the area). Weak layers and interfaces, and therefore metastability, can persist for weeks or months [Jamieson and Johnston, 2001], thus conserving hazard for a long period of time.
[9] In this scheme, two distinct types of disturbances are considered to be candidates for propagating the transition of energy states: (1) a transverse disturbance characterized by pure bending deformation in the slab and collapse of the adjacent weak layer perpendicular to the crack plane (see Figure 1) and (2) an in-plane disturbance characterized by both deformation and collapse parallel to the crack plane;
that is, a part of the slab is sliding down-slope on a weak interface between slab and basal layer and propagating the fracture on its perimeter. A fracture of this second type necessarily requires a slope to free energy. In this work we
examine disturbances of the first type, i.e., transverse disturbances.
3. Equation for Transverse Disturbance
[10] In this section the propagation of a firn quake or a whumpf as a transverse disturbance is analyzed. This situation is also characterized as rapid compressive loading [McClung, 1981].
[11] We consider a horizontal three-layer snow stratifica- tion consisting of a compact, nondeformable basal layer, a collapsible weak layer and a homogenous snow slab as top layer (see Figure 1). The displacements of any point in the slab in the directions of thex,yandzaxis are denoted by u1(x,y,t),u2(x,y,t) andu3(x,y,t) respectively.The absence of loading tensions in the xy plane in low-angle terrain allows us to examine disturbances with displacements parallel to thezaxis only; that is,u1andu2are identically zero. Further, we assume negligible viscous effects and associate the slab with a thin plate. The dynamical behavior of the slab is then governed by the differential equation for the transverse bending of thin plates [see, e.g.,Ge´radin and Rixen, 1997]:
D u3;xxxxþ2u3;xxyyþu3;yyyy
þrHu3;tt¼p3ðu;tÞ; ð1aÞ where r is the mean density andHis the thickness of the slab at location x. The right-hand term p3(u, t) is the resultant loading of the slab per unit surface. Thex,yandt subscripts indicate partial derivation. The coefficient D stands for the plate-bending stiffness (or flexural rigidity of plates) and depends on the elastic modulus E and the Poisson rationof the slab:
D¼ EH3
12 1ð n2Þ: ð1bÞ [12] Sufficiently far in the x direction from the trigger point, where the curvature of the disturbance in thexyplane becomes very small, lateral variation inybecomes negligi- ble and equation (1a) can be simplified to
D u3;xxxxðx;tÞ þrHu3;ttðx;tÞ ¼p3ðu;tÞ: ð2Þ Introducing g as the z component of the terrestrial acceleration and taking p3 equal to the weight of the slab per unit surface in disturbed area, where all derivates in x and t vanish, and p3 identically zero in undisturbed area, where the weight of the slab is exactly balanced by the reaction of the basal layer on the slab (see Figure 1), we get
D u3;xxxxðx;tÞ þrHu3;ttðx;tÞ ¼
rHg; in disturbed area 0; otherwise
8<
: :
ð3Þ We look for stationary solutions of equation (3) of the form u3(x, t) = u3(x ct), where c is the propagation velocity. For convenience we introduce the change of Figure 1. Characteristic data of a transverse disturbance
propagating through an elastic slab with velocity c. The weak layer collapses instantly near the crack front (atx0= 0, marked by an asterisk). The size of the weak layer is exaggerated. In general, h H. The total length of the collapse region isgl.
variable x0 = x ct and define the length l by the expression
lc¼ ffiffiffiffiffiffiffiffiffiffiffiffi pD=rH
: ð4Þ
Obviously l is a characteristic length of the disturbance and we refer to it as wavelength in the following. Equation (4) then becomes the ordinary differential equation
l2c2 d4
dx04u3ð Þ þx0 c2 d2
dx02u3ð Þ ¼x0
g; in disturbed area 0; otherwise 8<
: : ð5Þ
This is the stationary equation for fracture propagation of a transverse collapse.
[13] The roots of the characteristic polynomial of equa- tion (5) are both complex and equal to ±i/l. Hence the nontrivial solutions are of the form
u3ð Þ ¼ x0 g
2c2x02þC1þC2x0þC3cos x0 l
þC4sin x0 l
ð6aÞ
betweengl<x0< 0, wherex0= gldefines the contact point between the slab and the basal layer after the passage of the disturbance (see Figure 1). Outside this interval, where equation (5) is homogenous, we take u3(x0) identically zero or constant, but oscillating solutions in which we are not interested here are mathematically allowable. We search solutions satisfying the following boundary conditions:
u3ð Þ ¼0 0;
u3ðglÞ ¼h;
d
dx0u3ð Þ ¼0 0;
d
dx0u3ðglÞ ¼0;
d2
dx02u3ðglÞ ¼0:
ð6bÞ
The ‘‘crack settlement’’ h appearing in equation (6b) expresses the total downward displacement of the slab during the passage of the disturbance. It is taken positive in the upward direction and negative in downward direction (see Figure 1). A nonlinear algebraic set of five equations is obtained, from whichlcan be removed by substitution of equation (4). The remaining six variables are c,l,C1, C2, C3andC4. Therefore the equation set allows one degree of freedom. This remaining degree of freedom would normally be reduced by fracture mechanical considerations. However we will follow another lead by assuming that the collapse at the crack front is instant and involves no net force on the slab and the basal layer. Therefore instantly after the failure, the displacement of the slab at the crack front atx0= 0 obeys
free fall motion. In the moving coordinate system, this is expressed by
d2
dx02u3ð Þ ¼ g=c0 2: ð6cÞ This implies that the energy transition across the metastable state occurs at the slightest twitch, that is, we have neglected the dynamical effect of the activation energy for the state transition. The equation set is thus entirely defined. The solutions are as follows:
C1¼C3¼0;
C2¼gl c2 g
cosg1; C4¼ lC2;
c4¼ g 2h D
rH; l4¼ 2h
g D rH;
ð7aÞ
where it is recalled thathis negative in downward direction andg is determined by the implicit equation
g¼tang=2; g6¼0: ð7bÞ Only the smallest strictly positive root, g 2.331 rad, corresponds to a nontrivial, physically acceptable solution for a basal layer that takes no deformation.
[14] The solution of equation (5) contains some interest- ing properties. First, it is a localized, kink-shaped distur- bance that propagates without change of form, wavelength and velocity, as long as snowpack properties remain con- stant. According toDodd et al.[1982, pp. 9 – 11], these are properties of solitary waves. Second, we emphasize that the solution is not a wave in the usual sense with a finite Pointing vector. The disturbance is better described as a state transition. In reality, the disturbance does transport the activation energy of the state transition, but we assumed this to be very small and we neglected it in the model. Third, the stationary disturbance can virtually travel for infinite dis- tances through an appropriate, metastable snowpack. Zones without metastability are a barrier for propagation, but tunneling through small barriers up to the size of the order of l is not a priori impossible. Therefore small local anomalies in the stratification are unlikely to affect the propagation. Fourth, an interesting characteristic timescale related to the duration of the disturbance can be found in the ratio
l=c¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2j j=gh
p ; ð8Þ
which is independent of most stratification characteristics and equals the free-fall time over the crack settlement h.
Last, it is important to realize that the collapsible layer need not be a thin layer of buried surface hoar. If the disturbance triggers a sudden tighter packing of loosely packed aggregate particles in any buried snow layer [Truman, 1973], a necessary condition for fracture propagation is met.
[15] The components of the strain and the strain rate during the passage of the disturbance can be deduced from equations (6a) and (7a):
e13ð Þ ¼x0 1 2 g
c2x0þC2C2cos x0 l
_
e13ð Þ ¼x0 c 2
g c2C2
l sin x0 l
:
ð9Þ
All other components vanish. Especially the maximum strain rate at the crack front is simply given by the expression
_
e13ðx0¼0Þ ¼ g
2c: ð10Þ
[16] Finally, an interesting comparison can be drawn between fracture propagation velocity and the velocity of linear elastic waves by substitution of equation (1b) into the expression forcin equation (7a). This results in
c¼ ffiffiffiffiffiffiffiffiffiffiffiffi c0cs
p ; ð11Þ
where
c20¼ gH2
12j j h ð1nÞ c2s¼ E
2 1ð þnÞr: ð12Þ Here,csstands for the wave velocity of a shear linear-elastic wave in the snow slab above the weak layer and c0 is a coefficient with the dimension of a velocity.
[17] Truman[1973] has compared the propagation of the fracture in snow to the ‘‘domino effect.’’ This is true, not in a mechanical sense of snow particles toppling neighbor aggregates, but for its analogy with this very simple metastable system that also propagates a failure with con- stant speed in a homogenous arrangement.
4. Experimental Data
[18] Currently very little experimental data on crack propagation in snow are available and future efforts in this direction are desirable. In this section we describe the available data for testing equation (7).
4.1. Test Case 1: Johnson’s Bow Summit Experiment [19] The experiment mentioned in the introduction was carried out by Johnson at Bow Summit, Banff National Park, Canada [Johnson et al., 2005] on 19 February 2000.
The snow stratification exhibited a bonded slab resting on a 10 mm thick weak layer composed of surface hoar crystals ranging in size from 3 to 7 mm. The weak layer was estimated to be approximately 50 days old. The average density of the overlying slab was 190 kg/m3and its vertical thickness 0.40 m. The elastic modulus of the slab was estimated based on its density [Mellor, 1975] to be 1.0 MPa.
After a whumpf was successfully triggered with snow shoes, postmortem examination revealed a total vertical displacement of the slab between 1 mm [Johnson, 2001]
and 2 mm [Johnson et al., 2005]. The angle of the terrain in the area was near zero. These data are summarized in
Table 1. Johnson observed the velocity of the disturbance with a string of six geophones and reported a mean propagation velocity of 20.0 ± 2.0 ms1 between 4.8 m and 12.7 m from the trigger point [Johnson et al., 2005].
4.2. Test Case 2: Reports on Firn Quakes and Whumpfs
[20] Several qualitative and semiquantitative statements from reports on firn quakes and whumpfs can be used to test the model.
[21] First, firn quakes were described as wave-like dis- turbances that can travel for many kilometers apparently unaffected by local anomalies in the snow stratigraphy [Den Hartog, 1982], a property that seems to rule out any elastic longitudinal or shear waves as carriers of the fracture [Bohren and Betscha, 1974].
[22] Second, visual observations of firn quakes and whumpfs indicate a remarkably wide range of propagation velocities. In a snow-covered meadow with a thin overlying slab at most 0.15 to 0.30 m thick, a consistent whumpf propagating with rather constant 6 ms1 was observed in combination with a collapse of 10 to 20 mm of the snow surface on 19 March 1970 [Truman, 1973]. The failure was initiated by stamping on the snow cover and the velocity estimated from estimates of distance and time. In another situation,Bohren and Betscha [1974] report a quite rapid, audible ‘‘whumph’’ in combination with the collapse of depth hoar below 0.6 to 1.8 m of snow. A settlement of one or more centimeters was reported on this occasion. The velocity was said to be at least one order of magnitude greater than Truman’s observation, i.e., above 60 ms1. In a third situation in the Antarctic on 27 March 1958, described byDen Hartog [1982], a firn quake was experienced with an even much quicker propagation velocity close to the speed of sound in air. No signs of a collapsed firn horizon was found in a 3 m deep snow pit, suggesting that the fracture could have occurred below that depth.Den Hartog [1982] also suggests that the very fast March 1958 failure may have been caused by an entirely different phenomenon than the very slow one Truman observed in March 1970.
5. Discussion
[23] The results of Johnson’s Bow Summit experiment are compared with the predictions of equation (7) in Table 2, for a range of physically reasonable values of Poisson’s ratio [Mellor, 1975].
[24] In the low-angle terrain at Bow Summit, the propagation velocity of the whumpf predicted by equa- tions (7) results in 21.9 to 22.9 ms1, in good agreement with Johnson’s measured 20.0 ms1. As expected for this stratification, these velocities are well below the propa- Table 1. Input Data for the Model From Observations byJohnson [2001] at Bow Summit, Banff National Park, Canada
Data for Bow Summit FromJohnson et al.[2005] Range
Slope angle, deg 0
Accelerationg, ms2 9.81
zcomponent of slab thicknessH, m 0.40 ± 0.05
Crack settlementh, mm 1.5 ± 0.5
Mean slab densityr, kg m3 190 ± 20
Elastic modulusE, MPa 1.0 ± 0.4
gation velocities of elastic waves through the same slab.
For zero Poisson’s ratio, the remaining nonzero parame- ters were found as follows: l = 0.38 m, g = 2.33, C2 = 10.82 103 andC4= 4.14 103. Unfortunately no observation on l is available in this experiment, mainly because Johnson’s raw data are difficult to interpret in this respect [Johnson et al., 2005]. Interestingly however, the raw data show that the amplitude of the disturbance stayed constant in an interval between 4.7 and 12.7 m from the trigger point, which constitutes another argument in favor of a solitary-type fracture wave (the intensity of a radial elastic wave would decrease with increasing distance from the trigger point, whereas the solitary-type wave travels as a stable packet). Figure 2 shows the detailed functional solution of the disturbance. At the crack front atx0= 0, the displacement starts with a polynomial of second degree expressing free fall motion. However, the more the slab bends downward, the more elastic tension it builds up and the vertical motion is slowed down. Finally, at x0=gl, some 89 cm behind the crack front, the slab gets into contact with the fractured weak layer and basal layer (as previously stated, no bouncing back or oscillations are discussed here). Figures 3 and 4 display the strain and strain rate associated with the disturbance. The maximum strain rate attained in this case reached 0.22 s1, an unusually large value in snow characterizing the fracture as a brittle failure [Narita, 1980] and implying very low material strength of the disturbed weak layer [McClung, 1977; Narita, 1980;
Fukuzawa and Narita, 1992;Schweizer, 1998].
[25] Equation (10) actually shows that for the entire range of wave speeds that have been observed (from 6 to 330 ms1), a strain rate at the crack front much greater than 103s1is expected. At these rates snow behaves as brittle material [Narita, 1980], suggesting that brittle failure always occurs at the very front of the disturbance, regardless of snowpack properties.
[26] Equations (7a) show that the propagation velocity depends substantially on the value of h, H, r and E. In Figure 5, the dependence on H for typically observed combinations in instable snow is shown. Crack settlement was varied within a range between 1 mm and 25 mm, comprising all reported observations [Sorge, 1933; Den Hartog, 1982;Johnson et al., 2005], slab density was varied between 100 kgm3and 500 kgm3and elastic modulus E was estimated from the density [Mellor, 1975], ranging from 101to 103Mpa. The range of propagation velocities displayed in Figure 5 appears to be in agreement with the reported observations in test case 2. All situations are
explained by the model. Therefore contrary toDen Hartog [1982], we attribute all the observed velocities to the same phenomenon.
[27] Using the same combinations ofh,H,randEas in the previous paragraph, the coefficient c0 in equation (12) can be evaluated and its values are found to range between 2.8 ms1and 150 ms1for Poisson’s ratio below 0.4. In the case of Johnson’s Bow summit experiment, the values ofc0 range between 9.3 ms1and 12.0 ms1.
[28] Finally, due to the existence of stationary solutions, the model is in agreement with the observed ability of the disturbance to travel unaffectedly over large distances, as reported on several occasions in the Antarctic [Den Hartog, 1982]. This naturally supposes appropriate properties of the snowpack, as for example no dramatic increase of terrain angle and uninterrupted metastability.
[29] Before concluding this section, we revisit the hy- potheses we began with and compare them to our numerical application. Quite obviously, the small deformation assump- tion is valid, aslis much larger than hby a factor 102to 103. This justifies using equation (1a), which requires that the deformation is transverse and remains small. The effects of the processes neglected by the present model (viscous effects, thick plate effects) may account for differences between theory and experiment. Both these effects tend to slow down the propagation. The free fall assumption on the other hand tends to moderately speed up the propagation velocity. We expect the simplifying assumption of a non- deformable basal layer to produce sensible prediction in Table 2. Comparison of Model Results Predicted by Equation (7) and Experimental Results for Fracture Propagation Velocitya
Results for Bow Summit Range
Model Results
Calculated wavelengthl(equation (7)), m 0.38 ± 0.05
Total length of disturbancegl, m 0.89 ± 0.1
Calculated crack propagation velocityc(equation (7)), ms1 21.9 ± 4 forn= 0.0; 22.9 ± 4 forn= 0.4 Experimental Results From Johnson et al.[2005]
Measured crack velocityc, ms1 20.0 ± 2
Comparison With Elastic Waves
Calculated velocity of linear-elastic shear wavescs, ms1 51 forn= 0.0; 43 forn= 0.4
aExperimental results are fromJohnson et al.[2005]. A comparison of the fracture propagation velocity with linear-elastic shear wave velocitiescsis also shown. The results are given for two different values of Poisson’s ratio.
Figure 2. Graph of the displacement field of the kink- shaped solutionu3(x0) of equation (5). The example shown was computed with the stratification data reported by Johnson et al.[2005] for Bow Summit. Poisson’s ratio was taken zero.
many field cases, but situations with a soft basal layer may require to include oscillating solutions of equation (5) in the solution.
[30] The free fall assumption (6c) masks out some ongo- ing microstructural and fracture dynamical processes. The model thus provides no answers when studying the failure at the scale of snow particle aggregates. More comprehen- sive approaches must be employed for this type of inves- tigation, such as, for example, McClung [1979, 1981] or Louchet[2001]. The advantages of the present model with respect to comprehensive approaches are its simplicity, its requirement of field measurable input only (except for the elastic modulus, which can be estimated from density however) and its precise statements that can be tested in field experiments. This makes it especially suitable for field studies and practical application.
6. Summary and Conclusions
[31] An analytical model for fracture propagation in a partly collapsible snow stratification was proposed and applied to whumpfs and firn quakes. The model provides a framework for calculating a transverse disturbance re- sponsible for fracture propagation in low-angle terrain.
[32] A metastable snow stratification is regarded as a precondition for crack propagation. Metastability is a con- sequence of upper layers of snow storing gravitational potential energy that can be suddenly released by transverse
or in-plane sliding collapse of a weak layer or interface in a self-sustaining process.
[33] The solutions of the equation for fracture propaga- tion exhibit properties of solitary waves. It is a combination of properties of both the slab and the weak layer that determines the propagation velocity. The adherence of the slab to its basal layer can be destroyed during the passage of the disturbance, when the weak layer or interface collapses due to the very high strain rate generated by the disturbance.
Indeed, the maximum strain rates attain overcritical values at the very crack front, suggesting that the weak layer undergoes brittle failure during collapse and looses all shear strength.
[34] Although suitable data for testing the theory are still scarce at present, numerical results of the theory were compared with the available experiments and observations.
The accordance is very satisfying (Bow Summit experi- ment). Indeed, the difference between computed and mea- sured propagation velocity is largely within the uncertainty.
The model is in agreement with the observed ability of the disturbance to travel unaffectedly over large distances, as reported on several occasions in the Antarctic. The theory is further in agreement with the remarkably wide range of velocities (from 6 to approximately 330 ms1) reported in different loading situations.
Notation
c fracture propagation velocity inxyplane, ms1. c0 coefficient, ms1.
Figure 3. Graph of the strain fielde13(x,t) associated with the Bow Summit solution (see Figure 2). The peak strain is on the order of 103near x0= l.
Figure 4. Graph of the strain rate field associated with the Bow Summit solution (see Figure 2). The peak strain rate equals 0.22 s1at the crack front.
Figure 5. Range of propagation velocities for transverse disturbances in low-angle terrain (shaded area) for typically observed values of slab thickness (see text). Crack settlement was taken between 1.0 mm < h < 25 mm, according to reported observations. The error boxes represent the propagation velocities measured at Bow Summit and reported in test case 2. Abbreviations are as follows: T, Truman [1973]; J, Johnson et al. [2000, 2005]; BB, Bohren and Betscha [1974]; DH, Den Hartog [1982].
cs wave velocity of linear elastic shear wave, ms1. D plate-bending stiffness of the slab, Nm.
e13 strain component in slab, dimensionless.
E Young’s elastic modulus, Nm2.
g zcomponent of terrestrial acceleration, ms2. h crack settlement, i.e., total downward displacement, m.
H perpendicular slab thickness, m.
p loading of the slab per unit surface, Nm2.
u3 displacement of the slab in the direction of thezaxis, m.
gl distance between crack front and contact point after the collapse (see Figure 1), m.
l characteristic wavelength of the disturbance, m.
n Poisson ratio for linear-elastic body, dimensionless.
r mean density of the slab, kg m3.
[35] Acknowledgments. I would like to thank Jakob Rhyner for his advice in developing the theory and the reviewers for their constructive remarks. Several colleagues at SLF, especially Juerg Schweizer and Martin Jonsson, contributed with many useful discussions.
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J. Heierli, WSL Swiss Federal Institute for Snow and Avalanche Research, Flueelastrasse 11, Davos Dorf, GR 7620, Switzerland. (heierli@slf.ch)
