These authors contributed equally to this work.

Dry-snow slab avalanche release is preceded by a fracture process within the snowpack. Recognizing weak-layer collapse as an integral part of the fracture process is crucial and explains phenomena such as

In the first part of this two-part series we introduce a closed-form analytical model of a snowpack accounting for the deformable layer. Despite the importance of persistent weak layers for slab avalanche release, no simple analytical model considering weak-layer deformations is available. The proposed model provides deformations of the snow slab, weak-layer stresses and energy release rates of cracks within the weak layer. It generally applies to skier-loaded slopes as well as stability tests such as the propagation saw test. A validation with a numerical reference model shows very good agreement of the stress and energy release rate results in several parametric studies including analyses of the bridging effect and slope angle dependence. The proposed model is used to analyze 93 propagation saw tests. Computed weak-layer fracture toughness values are physically meaningful and in excellent agreement with finite element analyses.

In the second part of the series

Dry-snow slab avalanches can release when a persistent weak layer of, for example, surface hoar or depth hoar breaks (see the well-known image of a partially collapsed weak layer by

The earliest approaches to snowpack stability were so-called stability indices. They consider snowpack loading owing to the weight of the snow slab and owing to additional loading by a skier

Weak layer of buried surface hoar. Left of the vertical slab fracture the weak layer has collapsed, whereas on the right-hand side the porous weak layer is still intact. Reprinted from the Journal of Glaciology

To evaluate stability indices or fracture mechanics criteria, a model of the stress distribution within the snowpack and especially the weak layer is needed. Using the exact solution for a concentrated load on a homogeneous semi-infinite plate,

In the first part of this two-part contribution, a novel modeling approach for the description of weak-layer failure is given. It aims at providing a model that fully accounts for the weak layer's effect on deformations and load transfer and solutions to the mixed-mode energy release rates of cracks within the weak layer. The model is validated using finite element analyses and field experiments. In the second part we propose a new failure criterion that physically links stability indices and fracture mechanics models. Here, the necessary distinction of the strength of materials and fracture mechanics approaches is highlighted and discussed.

Deformation, stresses and consequently the energy release rate of cracks within the weak layer are controlled by loading and the complete stratigraphy of the snowpack. Deformations of the slab and in particular of the weak layer must be described in a sufficiently accurate manner. The slab is loaded in local bending and stretching, which we account for using beam and rod kinematics. As in the analysis by

Snowpack modeled

Consider the snowpack model on an inclined slope of angle

Timoshenko kinematics for the beam allow for shear deformation. Initially plane beam cross sections may rotate by an angle

Enforcing equilibrium of forces and moments and using the laws of elasticity allows for deriving a set of ordinary differential equations (ODEs) with constant coefficients that describe the deformation of the snow slab (see Appendix

Snowpack configurations assembled from beam segments with boundary and transmission conditions:

The general solution of ODE (

For regions without the elastic foundation, e.g., above a crack or when modeling propagation saw tests (PSTs), we obtain

The present model readily applies to an inclined skier-loaded snowpack with or without a crack in the weak layer as well as to propagation saw tests. For this purpose, considered snowpack configurations must be assembled from general solutions of beam segments with or without the elastic foundation given above. The free constants are determined from boundary and transmission conditions.

To model a propagation saw test two beam segments are required as shown in Fig.

In skier-loaded snowpacks (Fig.

The boundary and transmission conditions for the respective load case provide a linear system of equations with up to 24 unknown constants. The system can be solved easily using any mathematical toolbox. Closed-form solutions for

Graphical representation of the mode I crack opening integral. An uncracked snowpack can be represented by removing a part of the weak layer and applying virtual stresses to both the slab and the removed weak-layer segment such that they are deformed as in the original configuration (

Since the weak layer is represented as an elastic Winkler foundation, also known as the weak interface model

Fracture mechanics

In linear elastic materials the Griffith criterion may be reformulated using stress intensity factors

Within the framework of weak interface models the energy release rate corresponding to infinitesimal crack growth can be obtained from the local strain energy

We distinguish different crack opening modes. Mode I loading is a crack opening mode normal to the crack faces. Strictly speaking this comprises only symmetric deformations, which typically does not hold for cracks along material discontinuities with different stiffnesses. Mode II and III are shear crack modes with tangential displacements of the two crack faces. In this work only modes I and II are considered. Following the concept of anticracks

Loading which causes any combination of the three crack opening modes is called mixed-mode loading. Then mixed-mode failure criteria must be used to account for the different contributions to the total energy release and the required energy to extend a crack under mixed-mode conditions. This will be a subject in Part 2 of this work.

In order to extend the scope of fracture mechanics, which is only applicable to infinitesimal crack growth

The total potential energy difference

The present model provides slab displacements, weak-layer stresses and energy release rates for cracks within the weak layer as closed-form analytical expressions. In order to validate the model, stresses and energy release rates are compared against detailed finite element analyses (FEAs) and existing models. Results of several parametric studies are shown in detail.
Further, we compute the fracture toughness corresponding to critical cut lengths in propagation saw tests for a comprehensive set of 93 field experiments provided by

Material properties used throughout the present work.

Finite element model used for validation. Discretization of a snowpack with slab and weak layer. Cracks are introduced by removing all weak-layer elements. Skier loads are applied as vertical concentrated forces. Here, the case of a propagation saw test is shown as an example. The rigid base below the weak layer has a Young modulus of

Reference stress solutions and energy release rates are computed using the plane strain FEA model shown in Fig.

For the following considerations, the Young modulus is calculated from density

A weak-layer Young modulus of

With reference to

Normal and shear stresses owing to pure vertical and pure horizontal combined skier and slab weight loading, respectively. Comparison of the present model (blue) and the

Figure

Bridging effect evident in weak-layer normal stresses depending on the slab thickness. Peak stresses and the width of the skier-loaded snowpack changes with different slab thicknesses. Model results (solid lines) are compared against the reference solution (markers

The bridging effect of stiff slabs is studied in Fig.

The effective rigidity of a Timoshenko beam against vertical deflections is composed of bending stiffness

Effect of the weak-layer thickness on local weak-layer normal stresses. The weak-layer thickness changes the size of the skier-load-affected part of the weak layer and also the peak value of the stress. Thicker, more compliant weak layers distribute the load on a larger area and lead to lower peak stresses. Model results (solid lines) are compared against the reference solution (markers

Figure

Total, mode I and mode II energy release rates in propagation saw tests of different slope angles

Let us consider the energy release rate solution. Figure

Mixed-mode energy release rates of propagation saw test (PST) configurations. The mode I (collapse) and mode II (shear) contributions of the energy release rates of a cut length

Energy release rate

Impact of model assumptions on slab deformations.

Boundary conditions in PST experiments:

In Fig.

Fracture toughness

Figure

PST experiments can be conducted in one of two ways as depicted in Fig.

Figure

Predictions of the present model are found within a narrow range around the one-to-one line, indicating excellent agreement (

The values of the obtained fracture toughness are within 3 orders of magnitude. A total of 61 % of the values are within 1 order of magnitude around the median value of

This maximum is obtained for the following configuration:

Incremental energy release rate of cracks below a skier for three different slab thicknesses. The slab is locally loaded by a skier and by the weight load of the slab. The slab is modeled to have a length of 25 times the slab thickness, guaranteeing vanishing boundary effects. Model parameters other than

The previously given results of the energy release rate as given by the model show differential energy release rates

The presented closed-form analytical model of the snowpack contains two different levels of abstraction. The first level is the treatment of the snowpack as a linear elastic continuum. This is common for models of skier-triggered avalanches

The second layer of abstraction is the simplification of heterogeneous media to continua and engineering structures like elastic foundations or deformable rods and beams. If chosen correctly, accurate representations of the deformation and stress fields within the continua can be recovered with such elements of structural analysis, which are well established in civil and mechanical engineering

The studies shown in Figs.

The energy release rate of cracks within the weak layer is studied in detail in the validation studies shown in Figs.

The effect of the slab thickness on the energy release rate obtained for PST experiments is shown in Fig.

As pointed out by

Increasing the thickness of the weak layer while keeping all other geometry and material properties constant increases its compliance. That is, Heierli's assumption of rigidity leads to increasing discrepancies as the weak-layer thickness increases. This is reflected in Fig.

The analysis of 93 PST data points (Fig.

In the framework of continuum mechanics the PST must be considered a fracture mechanical experiment aimed at identifying the fracture toughness. The fracture toughness is the energy required to form a new surface on the idealized plane of fracture. This energy comprises the dissipative processes at the microscale. These processes are very different between crack growth under tension and under compression with a collapse of the weak layer. In the latter case local dissipative damage processes are much more pronounced, leading to a significantly higher value of the (effective) fracture toughness in compression

In their study

As pointed out by

Figure

The present model does not account for layering of the slab above the weak layer. It is well known that the layering can have a significant effect

Penetration depths leading to forces acting not on top of the slab but well within the slab are not considered at this point. To understand loading scenarios where the penetration depth is high, e.g., because of a soft top layer of snow or a highly localized load (recreationist by foot or a snowmobile), further modeling and experimental effort is needed.

In this work, interaction of shear and compression is limited. Slab extension causes only tangential and slab deflection causes only normal weak-layer deformations. This modeling strategy can be pictured as weak-layer springs attached to the midsurface of the beam. With this simplification the governing system of equations is uncoupled and can be solved independently for tangential and normal displacements, respectively. Of course, the weak layer does not interact with the midsurface of the slab but with its bottom side. Tangential displacements on the slab bottom side are caused by both slab extension and the rotation of beam cross sections, which in turn depends on the slab deflection. Hence, the rotation of beam cross sections couples tangential and normal displacements. This will increase the solution effort but simultaneously provide improved accuracy, in particular concerning weak-layer shear stresses and mode II energy release rates (see, e.g., Fig.

By considering a deformable weak layer the present work provides a simple but comprehensive closed-form analytical model for snowpack deformations, weak-layer stresses and energy release rates of cracks within weak layers:

The model applies to skier-loaded slopes as well as PST experiments.

Providing closed-form solutions, the present analysis framework is highly efficient and evaluates in real time.

Comparisons with numerical reference solutions indicate good agreement of weak-layer stress and energy release rates.

The model renders physical effects such as the bridging effect of thick slabs, the influence of slope inclination and most importantly the impact of weak-layer compliance. Evaluating a comprehensive set of 93 PSTs, fracture toughnesses of a multitude of weak layers are analyzed and discussed.

Providing weak-layer stress and energy release rates of cracks within the weak layer, the present framework allows for evaluating comprehensive criteria for skier-triggered crack nucleation and crack propagation in the weak layer, which will be discussed in Part 2 of this two-part work.

The set of differential equations governing the present problem can be derived from the equilibrium of forces and moments of a beam element and elasticity laws for the bending moment

Preliminary studies showed that two-parameter foundation models like Pasternak or Vlazov foundations

The ODE governing horizontal displacements, i.e., the deformations of an elastically bedded rod, is obtained by inserting the derivative of Eq. (

With normal and shear loading

The analysis code of both the modeling framework in Part 1 and the mixed-mode failure criterion based on this framework is available under

PLR and PW designed, conceived and developed the model. PW drafted the paper, and PLR carried out the numerical reference analysis. Both authors contributed equally to the presented model and the final version of the paper.

The authors declare that they have no conflict of interest.

We thank Johan Gaume and Alec van Herwijnen for sharing their PST data set and for valuable discussions on snow slab models and characterization tests. We want to thank Ned Bair, Karl Birkeland and Bastian Bergfeld for fruitful discussions on snow physics, model assumptions and snowpack stability experiments. We thank Wilfried Becker for his support and Vera Lübke for studying the feasibility of different modeling approaches. We are grateful for the contributions of Michael Zaiser, the anonymous referee and editor Guillaume Chambon, who carefully reviewed the paper.

This research has been supported by the German Research Foundation and the Open Access Publishing Fund of Technische Universität Darmstadt.

This paper was edited by Guillaume Chambon and reviewed by Michael Zaiser and one anonymous referee.