Dry-snow slab avalanches are generally caused by a sequence of fracture processes including (1) failure initiation in a weak snow layer underlying a cohesive slab, (2) crack propagation within the weak layer and (3) tensile fracture through the slab which leads to its detachment. During the past decades, theoretical and experimental work has gradually led to a better understanding of the fracture process in snow involving the collapse of the structure in the weak layer during fracture. This now allows us to better model failure initiation and the onset of crack propagation, i.e., to estimate the critical length required for crack propagation. On the other hand, our understanding of dynamic crack propagation and fracture arrest propensity is still very limited.

To shed more light on this issue, we performed numerical propagation saw test (PST) experiments applying the discrete element (DE) method and compared the numerical results with field measurements based on particle tracking. The goal is to investigate the influence of weak layer failure and the mechanical properties of the slab on crack propagation and fracture arrest propensity. Crack propagation speeds and distances before fracture arrest were derived from the DE simulations for different snowpack configurations and mechanical properties. Then, in order to compare the numerical and experimental results, the slab mechanical properties (Young's modulus and strength) which are not measured in the field were derived from density. The simulations nicely reproduced the process of crack propagation observed in field PSTs. Finally, the mechanical processes at play were analyzed in depth which led to suggestions for minimum column length in field PSTs.

Dry-snow slab avalanches result from the failure of a weak snow layer
underlying cohesive slab layers. The local damage in the weak layer develops
into a crack which can expand if its size exceeds a critical length or if the
load exceeds a critical value. Finally, crack propagation leads to the
tensile fracture of the slab and ultimately, avalanche release

In this paper, numerical experiments of the propagation saw test (PST) are performed by applying the discrete element (DE) method which allows us to mimic the high porosity of snow. The goal is to investigate the influence of weak layer failure and the mechanical properties of the slab on crack propagation. In the first section, field data as well as the proposed model are presented. Then, crack propagation speed and distance before fracture arrest are derived from the DE simulations using the same method as for the field experiments (particle tracking). In a parametric analysis, we show the influence of single system parameters on the crack propagation speed and distance. Finally, the interdependence of snowpack properties is accounted for in order to compare numerical and experimental results and the mechanical processes leading to fracture arrest are analyzed.

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Since the winter of 2004–2005, we collected data from 121 PST experiments at
46 different sites in Canada, USA and Switzerland

Schematic drawing and picture of the propagation saw test (PST). The black dots are markers used for particle tracking
in order to measure the displacement of the slab. The column length is denoted

Temporal evolution of the measured vertical displacement

Discrete element (DE) modeling

The objective of the proposed approach is to use the discrete element method (DE) to study the dynamic phase of crack propagation in a weak snowpack layer below a cohesive slab. The DE method is adequate for our purpose because (1) no assumption needs to be made about where and how a crack forms and propagates, and (2) the model material is inherently discontinuous and well adapted to dynamic issues. We will show that this method allows us to capture all the main physical processes involved in the release of dry-snow slab avalanches, namely the complex mechanical behavior of the weak layer and the interplay between basal crack propagation, slab bending, and slab fracture.

However, an important preliminary issue to address concerns the scale of the
considered model. In the weak layer, we intend to represent, through a
simplified description, the particular collapsible and highly porous
microstructure of the snow in order to be able to reproduce the main
features of the failure envelope of this material. As will be shown, we
achieve this by using triangular shapes of centimeter size. To account for
the possible breakage of these elements, they consist of small cohesive
grains of size

To summarize, we contend that, unlike in other DE applications which are at
the scale of the microstructure (e.g.

The discrete element simulations were performed using the commercial software PFC2D (by Itasca), which
implements the original soft-contact algorithm described in

The simulated system (see Fig.

The loading is applied by gravity and by advancing a “saw” (in red on Fig.

Mechanical parameters used in the simulations for the contact law.

The cohesive contact law used in the
simulations is the PFC parallel bond model represented schematically in Fig.

Mechanical parameters used in the simulations for the cohesive law.

The cohesive component (in black in Fig.

As the two components are acting in parallel, the total stiffness of the
bonded contact is equal to the sum of the contact stiffness

Failure envelope of the modeled WL obtained from mixed-mode shear-compression loading tests. The angle represented next to the
data points is the slope angle,

The time step was computed
classically as a function of the grain properties according to

Snapshots of a PST numerical experiment.

First, the macroscopic properties of the slab have to be determined as a function of the microscopic properties of the bond. For the slab, bi-axial tests were carried out which allowed to validate that for a squared assembly, the macroscopic (bulk) Young's modulus depends on bond stiffness according to Eq. (4).

For the weak layer, similarly, simple loading tests were carried out to
compute the macroscopic failure criterion (mixed-mode shear-compression) of
the WL as a function of the bonds of WL grains

Then, PST simulations were performed. An illustration of a simulation result
highlighting the displacement wave of the slab is shown in Fig.

In order to determine the crack propagation speed, purely elastic simulations
(infinite tensile and shear strength of the bonds between slab grains) were
carried out. The propagation speed was computed using the same method as for
field PSTs by analyzing the vertical displacement wave of the slab

Temporal evolution of the modeled vertical displacement

Snapshot of a PST with fracture arrest due to tensile crack opening in the slab induced by slab bending.

The propagation distance was computed by taking into account the possible
failure of the slab by setting finite values to the tensile and shear
strength of the slab (

For the parametric analysis (Sect.

Table of the parameter values used for Figs.

The evolution of the vertical displacement

Between 0 and 0.1 s nothing happens, then as the saw advances, the vertical
displacement slowly increases. This phase corresponds to the bending of the
under-cut part of the slab. Then, for

Crack propagation speed

Crack propagation distance

For all the simulations carried out, the crack propagation speed varied
between 5 and 60 m s

Finally, the speed of the elastic waves in the slab (

Figure

The influence of the Young's modulus

Then, the influence of WL thickness

Figure

Crack propagation distance slightly decreases with slab density as shown in
Fig.

Whereas slab density

Finally, crack propagation distance decreases with WL strength
(Fig.

The results of the previous parametric analysis should be interpreted with
care since for snow, several of the system parameters are inter-related
leading to more complex interactions. For instance, the result about the
influence of Young's modulus on the propagation distance might seem
contradictory to avalanche observations. Indeed, taken as it is, this result
would imply that it is easier to trigger a tensile failure in stiff and thus
hard snow than in soft snow. Consequently, hard slabs would result in
smaller release areas than soft slabs which is clearly in contradiction with
avalanche observations

In order to compare our numerical model to PST field data, we selected two simulation cases to show the overall trend of the propagation speed and distance with density, rather than simulating precisely each of the PSTs individually (which are prone to some variability) by using the available initial conditions from the field.

In the following, we distinguish two simulation cases:

Case #1 corresponds to simulations with a constant slab thickness

Case #2 corresponds to a case with a slope angle

A similar choice was made by

Average slab thickness as a function of slab density for PST field data.

Our numerical results (Fig.

The crack propagation speed

Overall, both simulation cases #1 and #2 reproduce the magnitude of the
propagation speed

Furthermore, the simulations of case #2 were done for the same conditions of
failure initiation, i.e., the strength of the WL bonds was calibrated in order
to have the same critical length for the different densities. However, for
the experiments, the critical length generally increases with increasing
density due to the settlement which induces an increase of Young's modulus
and a strengthening of the WL

Finally, for a low slab density

The proportion between the number of experiments for which fracture arrest
was observed

The crack propagation distance

The experiments and the simulations confirm that dense and hard snow slabs are more prone to wide-spread crack propagation than soft slabs.

Evolution of the normal stress

In order to better understand the underlying mechanical processes of fracture
arrest in the slab, the normal stresses in the slab

Then, once the critical length is reached, the crack becomes
self-propagating. The crack length increase leads to an increase of the
tensile stresses in the slab. Note that the maximum tensile stress

In order to better understand why fracture arrest does not occur anymore for
high densities, as shown in Fig.

For low values of slab density (

This result highlights the limits of the static beam theory and thus the need to take into account dynamic effects when addressing fracture arrest propensity issues. Indeed, we suppose that the reason of this sudden decrease is due to the crack propagation speed which becomes higher as slab density increases and induces a loss of support in the slab where stresses do not have time to establish. In other words, the displacement of the slab due to gravity is too slow to establish a mechanical equilibrium between bending and gravity. For instance, if we assume that the crack would propagate at an infinite speed, then the tensile stresses in the slab would not increase after reaching the critical length. The maximum tensile stress in the slab would thus be the one obtained at the moment of the onset of crack propagation. Obviously, the propagation speed is not infinite but limits the establishment of the stresses in the slab.

Using the theoretical relationships for

Obviously, the density

In this study, a numerical model based on the discrete element method was developed in order to perform numerical PST simulations and study the mechanical processes involved. Despite the apparent simplicity of the proposed DE model and of the structure of the simulated WL, we were able to quantitatively address the issue of the dynamic phase of crack propagation as well as fracture arrest propensity and to reproduce PST field data.

First, a parametric analysis was conducted to study the influence of snowpack
properties on crack propagation speed and distance. It was shown that the
propagation speed increases with increasing slab density

In addition, it was shown that the tensile fracture of the slab always
starts from the top surface of the slab. The propagation distance

Furthermore, by accounting for the relation between the mechanical properties of the snowpack, the increase of crack propagation speed and distance with increasing slab density was well reproduced. The slight overestimation of the propagation speed for low densities might be due to the fact that, to compute the propagation speed, the slab was considered as purely elastic and possible plastic effects in the slab that might induce energy dissipation were disregarded.

The in-depth analysis of the mechanical processes involved
in fracture arrest showed that after a certain slab density value

In addition, interestingly, in very few simulations both fracture arrest by
tensile failure of the slab and full propagation was observed. In these
cases, a portion of the WL on the right-side of the slab tensile crack was
damaged over a sufficient length to exceed the critical length leading again
to crack propagation. This process repeated itself until the end of the
system leading to so-called “en-echelon” fractures

Concerning the limitations of the model, we recall that the triangular shape
of the WL structure is highly idealized and that more complex and more
realistic geometries might have an influence on the presented results. In the
future, the micro-structure of the WL could be derived from micro-tomographic
images

Moreover, we would like to recall that the crack propagation speed was
computed from the vertical displacement wave of the slab. However, for high
values of the slope angle

With regards to practical applications, the results of our study can help to
choose the size of the column length in field PSTs. Indeed, we showed that
the maximum length for which snowpack properties might affect the propagation
distance is around 2 m, in agreement with the study of

We proposed a new approach to characterize the dynamic phase of crack propagation in weak snowpack layers as well as fracture arrest propensity by means of numerical PST simulations based on the discrete element method with elastic-brittle bonded grains.

This model allowed us to compute the crack propagation speed from slab vertical displacement as a function of snowpack properties. Furthermore, crack propagation distance was computed by taking into account the tensile strength of the slab. A parametric analysis provided the crack propagation speed and distance as a function of the different snowpack properties. We showed that the propagation speed increases with increasing Young's modulus of the slab, slab depth, slab density and slope angle but decreases with increasing weak layer strength. The propagation distance decreases with increasing Young's modulus of the slab, slab density and weak layer thickness but increases with increasing slab tensile strength, slab depth, weak layer strength and slope angle.

Then, the existing relationship between slab thickness, Young's modulus and
tensile strength with density was implemented. Accounting for this
relationship, modeled propagation speed and distances were found in good
agreement with those obtained from field measurements with the propagation
saw test.
In particular, for densities ranging from 100 to 300 kg m

For slab layers denser than

In the future, an in-depth analysis of crack propagation speeds for large
slope angles will be carried out in order to distinguish the speed associated
with the collapse wave of the slab and the speed associated to its tangential
displacement. Finally, different and more complex structures for the WL will
also be implemented with the long-term objective to model the structure of
the WL directly from segmented micro-tomographic images

Propagation speed as a function of the vertical displacement threshold

The method to derive the crack propagation speed from the evolution of the
vertical displacement of the slab in field and simulated PSTs is the same as
that described in

We thank Pascal Hagenmuller, an anonymous reviewer and the associate editor Eric Larour for their valuable comments and remarks that helped us to improve our paper. J. Gaume was supported by a Swiss Government Excellence Scholarship and is grateful to the State Secretariat for Education, Research and Innovation SERI of the Swiss Government. Benjamin Reuter, Eric Knoff and Mark Staples assisted with field data collection. Edited by: E. Larour