Rapid and large deformations of snow are mainly controlled by grain rearrangements, which occur through the failure of cohesive bonds and the creation of new contacts. We exploit a granular description of snow to develop a discrete element model based on the full 3-D microstructure captured by microtomography. The model assumes that snow is composed of rigid grains interacting through localized contacts accounting for cohesion and friction. The geometry of the grains and of the intergranular bonding system are explicitly defined from microtomographic data using geometrical criteria based on curvature and contiguity. Single grains are represented as rigid clumps of spheres. The model is applied to different snow samples subjected to confined compression tests. A detailed sensitivity analysis shows that artifacts introduced by the modeling approach and the influence of numerical parameters are limited compared to variations due to the geometry of the microstructure. The model shows that the compression behavior of snow is mainly controlled by the density of the samples, but that deviations from a pure density parameterization are not insignificant during the first phase of deformation. In particular, the model correctly predicts that, for a given density, faceted crystals are less resistant to compression than rounded grains or decomposed snow. For larger compression strains, no clear differences between snow types are observed.

Knowledge of the mechanical properties of snow is required for many
applications such as avalanche risk forecasting, optimizing over-snow vehicle
traffic, quantifying loads on structures, etc.

Numerous studies have attempted to measure the macroscopic mechanical
behavior of different snow types under various loading conditions (strain
rate and type of experiment) and temperatures.

Models incorporating state variables describing microstructural features such
as grain and bond size were developed

Rapid and large deformations of snow are mainly controlled by grain
rearrangements, which are enabled by bond failure and creation. This type of
deformation is involved in the release of slab avalanches

In the present paper, we propose to model dry snow deformation at high
loading rates using the DEM approach based on the real 3-D microstructure of
snow directly captured by

Seven different

Description of the microtomographic images used in this study. All
images are cubic. Density and equivalent spherical radius were computed
directly from the binary images. The equivalent spherical radius

Example of different snow microstructural patterns used in this study:

In practice, the output of tomography is a binary image composed of air and a
continuous ice matrix, without clearly identifiable grains. The first step thus
consists in detecting grains in these binary images. Since the grains are not
well separated but rather sintered together, their definition is generally not
objective. In the present study, the algorithm developed by

The segmented grains are represented by sets of voxels. This description must
then be translated into discrete elements that can be handled by a DEM model.
In principle, clumping of basic geometrical units such as spheres

In the present study, a very simple technique based on non-overlapping
spheres was used to describe (or mesh) the snow grains into discrete
elements. Following grain segmentation, the resolution of the 3-D image was
reduced by an adjustable factor. Then all voxels at the interface between two
grains or between air and ice were replaced by a sphere with a diameter equal
to the side length of the voxel with the same center position
(Fig.

Granular description of the ice matrix. The binary
image

Discrete element modeling was performed with the Yade DEM code
(

In this section, the contact law between two spheres of radius

In this study, cohesion strengths in tension (

Considering now intergranular contacts, they are generally composed of
several sphere–sphere contacts as described above (Fig.

Behavior of a sphere–sphere contact

A sufficiently short time step d

To dissipate kinetic energy and avoid numerical instabilities, a numerical
damping (coefficient

The simulations presented in this paper were run on a desktop computer with
a single processor (2.7 GHz) and 16 Gb RAM. The typical computing time of
the simulation is on the order of 1 day. More precisely, the test shown in
Sect.

The boundary conditions were chosen to reproduce strain-controlled vertical
confined compression of snow. Hence, the samples were placed between two
horizontal plates discretized using spheres of the same size as the spheres
used to describe the grain shape (Fig.

The velocity of the upper plate

Boundary conditions used in the present study. The grains (medium
gray) are compressed between two plates composed of spheres (black): the
bottom plate is fixed and the top plate is moving down with a constant
velocity

First, a typical simulation performed with the sample of rounded grains
(s-RG1) is described to give an overview of the deformation mechanisms
occurring during compression loading. A sensitivity analysis of the model to
the DEM parameters and microstructure representation is then presented.
Lastly, the model is applied to all snow images described in
Table

The model parameters used for this first simulation are summarized in
Table

Model parameters used for the simulations presented in this paper.

After the initiation of the compression, the snow structure first undergoes
elastic deformation. The work of the loading is completely stored as elastic
energy at the bonds. No cohesive bonds are broken, nor are any new bonds created.
If the upper plate is moved back to its initial position, the stress decreases
to zero and the microstructure fully recovers its initial state with no residual
deformation or stress. As a consequence, a linear relationship between stress

When the strain increases (

In this phase, hardening of the structure is observed. The stress
progressively increases, following an exponential-like trend. This phase
starts when the grain packing has reached a certain density (here

Results of the model applied to sample s-RG1 (rounded grains) with
the parameters listed in Table

A specific sensitivity analysis was conducted for the following model
parameters (Table

Example of the discrete element representation of two grains but with different effective resolutions (diameters of spheres).

Simulated stress–strain curves for different sphere
sizes

Figure

The total number of grains in the sample is not absolute, but depends on the
chosen segmentation parameters

Simulated stress–strain curves for different microscopic friction
coefficients

With the value

In order to expedite the simulations, the density of ice

Computed stress–strain curves for different Young's
moduli

The REV can be defined as the smallest
fraction of the sample volume over which the measurement or simulation of a
given variable will yield a value representative of the macroscopic behavior.
We assessed the representativity of mechanical simulations performed on
volumes of about

Simulated stress–strain curves for different volume sizes on sample
s-RG1

The numerical compression experiments were applied to different snow samples
spanning different snow types (see Table

Figure

A remarkable result, shown in Fig.

For density values greater than 300 kg m

Simulated stress–strain curves

We discuss here the potential artifacts introduced by the model in the simulation of the mechanical behavior of snow under compression.

Let us recall that the model is composed of rigid grains that cannot deform.
Only the contacts between members of different grains can deform to account
for intergranular strain. With FE simulations

This phase is controlled by the progressive failure of cohesive bonds and the
friction between grains. Failures of intergranular bonds are determined by
the force distribution in the sphere–sphere interactions. According to the
sensitivity analysis, the model appears to be insensitive to the size of the
spheres used to discretize the shape of the grains, and to the number of
grains (in reasonable ranges). The friction between grains depends on the
roughness of the grain surfaces and on the microscopic friction coefficient.
The regular mesh used in the present study tends to create “bumpy” grain
surfaces (Fig.

In this phase, rearrangements of grains can no longer be accommodated within the pore space. The grains form a dense packing. Above a certain macroscopic stress level, we can expect high intra-granular stresses to develop, which could lead to ice grains breaking into smaller parts. The assumption of “unbreakable” grains thus becomes questionable. Moreover, the high intergranular stresses yield large overlaps between spheres compared to their radius. The assumption of rigid grains is no longer valid and the simulated behavior becomes very sensitive to the value chosen for the Young's modulus.

In summary, the approach retained in this study, consisting in modeling snow as a granular material, is certainly reasonable and promising when attempting to reproduce the brittle/frictional deformation phase but is inappropriate to simulate pure elasticity or high compaction phases.

We discuss here the first-order role played by density on the simulated
mechanical behavior. As explained in the introduction, density is often
described as an insufficient mechanical indicator, because of the substantial
scatter observed in property–density relations derived from direct
experimental measurements

Furthermore, the dominant role played by density in the mechanical behavior
simulated is probably also relative to the loading conditions considered. In
compression, both the mechanical properties and density are expected to
depend on the grain bonding system, thus promoting the existence of an
apparent relation between stress and density

This paper presents a novel approach to modeling the mechanical behavior of snow under large and rapid deformations based on the complete 3-D microstructure of the material. The geometry of the microstructure is directly translated into discrete elements by accounting for the shape of the grains and the initial bonding system of the ice matrix. The grains are rigid and the overall deformations are only due to the geometric rearrangements of grains made possible by bond failure. The sensitivity analysis of the model to its parameters showed that the effects of the microstructure geometry on the simulated mechanical response are not shadowed by numerical artifacts.

In this study, the model was used to reproduce the mechanical behavior of
snow under confined compression. The representative volume related to this
type of loading was estimated to be

To further investigate the influence of microstructure on snow mechanical
properties, this DEM model could straightforwardly be applied to other
loading conditions. In particular, investigating shear loading can be useful
in the context of slope stability and avalanche release. Another promising
prospect would consist in numerically reproducing the penetration of an
indenter in the snow microstructure, with the objective of better
understanding the signals delivered by micro-penetrometers, which are
routinely used for snow characterization

Funding by the VOR research network (Tomo_FL project) is acknowledged. We thank T. Mede for his preliminary work on the model and the scientists of 3SR Laboratory and ESRF ID19 beamline, where the 3-D images were obtained. We also thank the CEN staff, especially F. Flin and B. Lesaffre, for the data acquisition. Irstea and CNRM-GAME/CEN are part of Labex OSUG@2020 (Investissements d'Avenir, grant agreement ANR-10-LABX-0056). Irstea is member of Labex TEC21 (Investissements d'Avenir, grant agreement ANR-11-LABX-0030). We thank M. Hopkins and two anonymous reviewers for their constructive feedback on this work. Edited by: M. Schneebeli