These authors contributed equally to this work.

Using the analytical model presented in Part 1 of this two-part paper, a new conceptual understanding of anticrack nucleation in weak layers is proposed. To obtain a sufficient condition for onset of failure, two necessary conditions must be satisfied simultaneously: (i) the weak layer must be overloaded in terms of stress and (ii) the initiating crack must release enough energy for the formation of new surfaces. This so-called coupled criterion was proposed by

Analyses of skier-loaded snowpacks show the impact of slab thickness and slope angle on critical loading and crack initiation length. In the limit cases of very thick slabs and very steep slopes, we obtain natural avalanche release. A discussion of different mixed-mode stress and energy criteria reveals that a wrong choice of mixed-mode hypotheses can yield unphysical results. The effect of material parameters such as density and compliance on weak-layer collapse is illustrated.

The framework presented in this two-part series harnesses the efficiency of closed-form solutions to provide fast and physically sound predictions of critical snowpack loads using a new conceptual understanding of mixed-mode weak-layer failure. It emphasized the importance of both stress and energy in avalanche release.

To study the onset of weak-layer failure, fracture mechanics models have been proposed that extend classical stability indices. Classical fracture mechanics is restricted to the analysis of growth of existing cracks. For a crack to propagate, a sufficient energy release is required to overcome the energy barrier for crack growth that originates, e.g., from surface energy and dissipative processes. Fracture mechanics has been applied to the analysis of weak-layer shear cracks

Here, anticrack refers to the extension of a collapse

In recent years fracture mechanics approaches were used and improved by many researchers

In order to characterize the resistance against crack propagation, a new field test, the propagation saw test (PST), was developed. It was first mentioned by

To link stress-based stability indices and fracture mechanics criteria, combined models were proposed.

Besides mechanical models that aim for closed-form analytical descriptions of the processes within the snowpack, numerical models were developed to study nonlinear failure processes within the snowpack.

The present work aims at providing a physical explanation for skier-triggered anticrack nucleation in weak layers. For this purpose we propose a unified failure criterion that directly links strength of materials and fracture mechanics. The criterion accounts for mixed-mode shear and compression loading. We employ closed-form analytical expressions for weak-layer stresses and energy release rates of cracks within the weak layer given in Part 1 of this series

Dry-snow slab avalanche release is typically preceded by the nucleation of an anticrack within the weak layer. As discussed by many researchers, skier-triggered weak-layer collapse is a problem of both strength of materials and fracture mechanics

Consider the problem of edge crack nucleation in four-point bending tests. Figure

In addition, statistical size effects may play a role as well. As shown by

Size effect of wood beams in 299 self-similar four-point bending experiments. Experimental results reported by

As a second example, consider a homogeneous isotropic bar subjected to a critical strain

In order to resolve the contradictions in the above examples, let us reconsider the four-point bending problem. Instead of examining local processes only, i.e., considering local stresses and expecting infinitesimal crack growth from the edge, let us assume the sudden formation of a finite-sized crack. The concept is known as finite fracture mechanics (FFM) and was proposed by

Let us now consider such finite cracks in thickness direction on the tensile side of the beam shown in Fig.

The above examples show that (i) fracture processes are governed not by one exclusive but by two conditions simultaneously even if one often hides the other and (ii) strength and toughness are independent fundamental material properties and one cannot be computed from the other.

The concept of finite fracture mechanics is best understood considering the limitations of classical fracture mechanics. The differential energy release rate

See Sect. 2.5 in Part 1

Coupled stress and energy criterion for the case of a skier-loaded snowpack. On the left the condition of a locally overloaded weak layer and on the right the second condition of sufficient energy release rate are shown. The stress condition has the effect of an upper bound on possible nucleated cracks, whereas the energy condition provides a lower bound on the crack lengths as short cracks do not release sufficient energy.

Besides the critical loading, the size of the initiating finite crack

Requiring the simultaneous satisfaction of both the energy and the stress condition,

Equation (

This general observation also applies to skier-triggered weak-layer collapse. Skier loading induces a stress concentration within the weak layer that may allow for the nucleation of finite-sized anticracks provided the given loading satisfies both the stress and the energy condition. The subsequent stability of this initial anticrack is governed by the energy condition alone.

For the application of the general coupled stress and energy criterion, Eq. (

Failure of shear- and compression-loaded geological materials such as soil and rock is often modeled using the Mohr–Coulomb strength criterion:

Smooth (present) and

Despite considering the influence of normal stress, the Mohr–Coulomb criterion, Eq. (

In order to simplify the mathematical formulation of the cap and to provide a smooth envelope, we propose a new effective weak-layer shear strength:

Mixed-mode energy criteria describe the interaction of crack opening modes I, II and III. Mode I corresponds to crack opening normal to the crack faces, which comprises both tearing and collapse each associated with a distinct fracture toughness,

For the interaction of compressive mode I and mode II loading the following simple general mixed-mode energy criterion,

Other mixed-mode criteria typically used in engineering applications such as the classical criteria by

As discussed in Sect.

Figure

Solution of the coupled stress and energy criterion for anticrack nucleation. The model presented in the first part of this work provides required stresses and energy release rates. Using the given closed-form analytical solutions, the minimum critical skier force satisfying both conditions can be identified efficiently by employing optimization procedures.

The present implementation of the coupled criterion provided as a Supplement

In the following, we use the mechanical model derived in Part 1

It is important to point out that the finite crack length

The second of the two necessary conditions for crack nucleation that are coupled in the coupled criterion (Eq.

The relation between incremental and differential energy release rates is given by

We consider the slope angle

For static skier loading the local load acting on the snow surface is

Material properties used in parametric studies of the finite fracture mechanics criterion.

Effect of the slope angle

Figure

Critical loading of a snowpack without assumed defects and the size of initiated anticrack as a function of slab thickness. Thicker slabs transfer concentrated loads less localized into the weak layer, allowing for larger point loads. Above a certain thickness, failure is dominated by the slab's own weight, reducing admissible additional loads until self-release occurs. The finite anticrack length increases continuously with increasing slab thickness.

The slab thickness directly affects the critical load. Figure

Effect of slab density

Figure

The coupled stress and energy criterion accounts for shear and compressive failure. In the following the effect of the mixed-mode criteria for stress and energy will be discussed.

Comparison of mixed-mode energy criteria for a defect-free snowpack loaded by a point load and the weight of the slab above the weak layer.

The choice of the mixed-mode fracture envelope and the ratio of the mode I and mode II fracture toughness is studied in Fig.

The effect of the criterion for the interaction of shear and normal stress in the weak layer is studied in Fig.

Comparison of mixed-mode stress criteria for a defect-free snowpack with slab thickness

In the following, results of the parametric studies given above are discussed in order to elucidate basic features of the present failure criterion.

The coupled stress and energy criterion employed in the present work uses linear elastic solutions provided in Part 1 of this work

Local overloading (stress exceeding strength) and sufficient energy release are only necessary conditions of failure

This was also addressed by

The present model extends the concept of anticracks by combining strength and energy as coupled conditions for the nucleation of anticracks. Physical interaction criteria of shear and normal stress as well as the mixed-mode energy release are covered. The anticrack model by Heierli extended the understanding of avalanche release and included the remote triggering of avalanches and whumpf sounds for which shear failure models cannot provide physical explanations. However, since the model only considers the weak layer's fracture toughness but not its compliance, important parameter dependences like the effect of slope angles contradict observations

The effect of slope angle

Figure

The effect of slab density

The mixed-mode law for the interaction of energy release rates of compressive (mode I) and shear (mode II) deformation of the initiated crack is studied in Fig.

Figure

The present work studies the onset of weak-layer failure. For a subsequent release of an avalanche, the growth of this crack and the trough-thickness fracture of the slab release are decisive as well.
Crack growth is directly controlled by the stability of the initiated and subsequently growing crack. The case of infinitesimal crack growth is covered by Griffith's criterion of linear elastic fracture mechanics (LEFM) and depends only on differential energy release rates in compression and shear. The present model also provides these quantities as an outcome of the model in Part 1, and the stability of the initiated cracks can be assessed in conjunction with the mixed-mode energy criterion (Eq.

Slab failure is typically induced by a combination of local bending and tension loading of the slab leading to slab fracture

Since, the propagation of existing cracks is purely energy-controlled

Effects of the slab and the fracture behavior of the weak layer will always interact, causing a complex failure behavior. Good models may allow for separating effects and, thus, allow for a better understanding of field experiments. For instance, the critical cut length in PST experiments may correlate with slab thickness but does not directly depend on it. Further,

The present model is based on closed-form equations of the slab and the weak-layer displacement fields. The proposed failure criterion for nucleation of anticrack makes use of this model and solves the implicit equations of the coupled criterion with high efficiency. Since the computational effort is much smaller than for numerical models, the present model can be used readily in large parameter studies or uncertainty quantification analyses.

A novel criterion for anticrack nucleation has been proposed on the basis of the closed-form analytical solution proposed in Part 1 of this work.

The criterion implicitly links a stress criterion for local overloading of the weak layer with a global fracture mechanics criterion of the energy balance of crack initiation.

It is shown that in order to study weak-layer failure the interaction of shear and compression stresses and mixed-mode energy release rates must be considered. Failure is governed by both strength and fracture toughness properties of the weak layer.

Parametric studies show that the proposed failure criterion is able to correctly render physical effects observed in slab avalanche release or field tests.

The model can be the basis for further analysis of the mixed fracture of the weak layer, the propagation of weak-layer failure and the failure of the slab above the weak layer.

The main limitations of the present model are the assumptions of a homogeneous slab, the isotropy of the weak layer, and the missing coupling of shear and normal displacements. Future works should address these points and investigate the required material properties, in particular elastic properties of the weak layer and the governing fracture and failure envelopes.

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

Both authors defined the scope of the work and developed the present failure model together. PW provided most of the introduction and the discussion. PLR conducted the parametric studies. PLR and PW wrote the final paper with equal contribution.

The authors declare that they have no conflict of interest.

We would like to thank Alec van Herwijnen and Johan Gaume for detailed discussions of the present work and the current understanding of the physics of slab avalanche release. We thank Karl Birkeland, Bastian Bergfeld and Ned Bair for the interesting exchange on snow stability and modeling of weak-layer failure. We are grateful for the contributions of the two referees, Michael Zaiser and Jürg Schweizer, 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 Jürg Schweizer and Michael Zaiser.