Dynamic Process of Dry Snow Slab Avalanche Formation: Theory, Experiment and Numerical Simulation

Abstract

Snow avalanches occur in snow-covered highland mountains and represent one of the most significant natural hazards pertaining to the field of geoscience. Although some insight into the formation of avalanches has been provided, a comprehensive overview or critical review of the latest research is currently lacking. This paper reviews recent advances on the formation process of dry slab avalanches and provides a guiding framework for further research. The formation of avalanches is the consequence of a series of fracture processes in the snowpack, which is usually induced by the failure of a weak layer underlying a snow slab layer. The parameters at each stage of avalanches’ formation are reviewed from theoretical, experimental and simulation perspectives. In terms of the onset of crack propagation, the understanding of the mechanical process has gone through a transition from shear theory, to the anticrack model and supershear. The critical length shows divergent trends with snowpack parameters and slope angles, and there is a lack of consensus in different models. The specific fracture energy is also an essential component in determining fracture propagation. Within cracks’ dynamic propagation, the crack propagation speed includes both the sub-Rayleigh regime and supershear. The crack speed exceeds the shear wave speed in the supershear mode. When the crack propagation reaches a specific distance, the slab undergoes a tensile fracture and the cracking’s arrest. The numerical simulation allows a complete reproduction of the initial failure, the crack’s dynamic propagation and slab fracture. In the future, a unified model is necessary through refining the formative mechanism and integrating it with the avalanche flow. This work offers a comprehensive understanding of the mechanics of the formation and release of avalanches, useful for both modelers and experimentalists.

1. Introduction

Snow is one of the most active natural elements on Earth. It serves as a vital source of material for polar ice caps and alpine glaciers [1,2,3]. Similarly, snow is an important material source for avalanches on land [4,5]. The microstructure of snow is on a scale of 10⁻⁴ m [6], and it is bound together by ice crystal particles. The mapping exhibits porosity and high compressibility at macroscopic scales [7,8,9]. The low strength and high deformability of snow compared to other materials results in a snowpack that is susceptible to destabilisation and avalanche triggering [9,10,11]. The formation and release of avalanches is a crucial process in avalanche science and the primary stage of avalanches. Avalanches occur in areas with snow cover or short-term snowfall and have significant impacts on vegetation, infrastructure and public safety [12,13,14,15,16]. Therefore, there are important roles for the study of their formation in the scientific prevention and early warning of avalanches. The main scientific topics of avalanche research include avalanche geography, avalanche formation and avalanche dynamics [17]. We consider the mechanics of avalanches’ formation in this paper, which is a critical stage in the initial triggering of avalanches.

The classification of avalanches is dependent on the manner in which they are triggered. There are various types of avalanches that can be classified according to the conditions under which they occur. This includes the dry slab avalanche, the wet snow avalanche, the powder avalanche, the loose snow avalanche, the ice avalanche, the slush avalanche and the gliding avalanche [13,18]. In particular, the dry snow slab avalanche represents the most dangerous type, accounting for more than 90% of all avalanche fatalities [6], and it is the most common avalanche. The dry snow slab avalanche has become one of the main studied avalanche types [19,20,21]; thus, the formation mechanism of dry slab avalanches is the primary focus of this review. Dry slab avalanches’ formation and release are characterised by a series of fracture processes. The snowpack is mostly a non-uniform anisotropic structure consisting of many layers with different mechanical properties [22,23,24]. The snow layer in dry slab avalanches is comprised of three layers (Figure 1); overlying the top layer is the slab, which is characterised by denseness and isotropy, and which is the main source of avalanche material. The middle with a higher porosity is the weak layer. In comparison, the weak layer displays inferior mechanical structural and fragile properties, and it is more susceptible to destructive failure due to external disturbances. The bottom layer is the substrate, which is characterised by rigidity and provides a supporting role. The failure of the weak layer below the cohesive slab is a necessary and insufficient condition for the formation of dry snow slab avalanches [21,25,26]. The weak layer has a low elastic modulus and strength and also has a strongly anisotropic structure [10,27,28]. As a result, weak layer failure is the primary factor in the formation of dry slab avalanches.

The primary factors involved in microcrack nucleation and avalanches’ formation within the snowpack are snowpack properties, snowfall and wind fields, temperature and humidity. These factors facilitate snow layers’ metamorphosis and the nucleation of microcracks at different loading rates, thus triggering avalanches [6,7]. The formation and release of dry snow slab avalanches is essentially a snow fracture process [6,19,30,31,32,33,34]. The initial failure occurs in the weak layer with a high porosity, resulting in the dynamic self-sustained propagation of mixed-mode and quasi-brittle cracks. If the terrain angle is larger than the snowpack friction angle, the slab eventually slides and releases [20,35,36,37]. Typical terrain angles for the formation of avalanches are between 30 and 50°, with avalanches more frequent in terrain with a 38° angle. The initial failure is the first stage of the formation of avalanches (Figure 1a) and generally starts at the microscopic scale [38,39], when the snowpack layer is defective at the microscopic scale, i.e., the snow particle adhesive bonds are broken. As a result of continuous loading, the defects gradually grow to the macroscopic scale, then the initial failure sprouts. The primary distinction between natural and artificial triggering lies in the initial phase. Once the macroscopic crack has reached a critical length, it will begin to propagate rapidly [20,25]. Hence the investigation of the critical crack length is also a pivotal physical quantity for the formation of avalanches (Figure 1b). Most of the experimental results showed critical lengths between 20 and 40 cm [40]. In natural avalanche release, however, the critical crack lengths are probably much longer (1–10 m) [41]. In addition, the energy release rate and specific fracture energy are crucial variables in the investigation of the onset of crack propagation [42]. During cracks’ dynamic propagation, the crack speed is the quantity most concerned (Figure 1c). Johnson et al. [43] first measured a crack propagation speed magnitude of about 20 m/s by seismic sensors. Recent experiments have measured an average crack propagation velocity in the range of 30–54 m/s [44,45]. The speed measurements also display scale-independence [45]. However, the supershear model indicates that the crack propagation speed can exceed one hundred metres per second [37,46,47], which reasonably explains the results of the experimental observations [48]. The final stage of the formation of avalanches is slab fracture and crack arrest (Figure 1d) [6,21,29]. The dynamic crack propagation and crack arrest in the weak layer are related to the propagation distance and ultimately determine the avalanche size [29,40]. Crack arrest is the stopping of the crack’s self-sustained expansion in the weak layer. The slab is subject to tensile force due to gravity, resulting in its stretching and fracturing. This causes the slab to release and subsequently slide. The snow slab fracture is caused by spatial variability in the snowpack. When the local stress is strong or the slab is thinner, the energy required for crack expansion in the weak layer may be greater than the energy released during crack propagation [49]. At this moment, the crack expansion stops, and the slab stretches and ruptures. It was observed that the propagation distance increased in accordance with the tensile strength and density of the snow slab. This indicates that the greater the tensile strength and density of the snow slab, the greater the distance it was able to expand. Conversely, softer slabs were more susceptible to fracture [21,50]. During sliding, snow slabs collide and break up and entrain the surface snow cover, resulting in the formation of large-scale avalanches [51,52]. Although avalanches are a large-scale dynamic process, the fracturing process that precedes avalanches’ release occurs on smaller scales, ranging from a few centimetres to metres [53]. The four stages of the formation of avalanches exert an influence on one another, with each playing an important role in the overall process. Therefore, a quantitative study of each process is fundamental to understanding the formation of avalanches. Even though we currently understand this process, however, there is a lack of a complete overview of avalanches’ development and a collation of the latest research outcomes. Furthermore, there is also a lack of discussion of existing debates and conflicting views about the formation of avalanches.

In this work, we detailed the dynamics of the formation of avalanches in different stages and corresponding physical quantities. In particular, each stage is described in terms of theoretical analysis, field experiments and numerical simulations. Next, in Section 2 we develop a review of the amount of attention paid to each stage of the formation of avalanches from theoretical and experimental developments and a few numerical simulation results for comparison. It includes the critical length and energy release rate of the onset of crack propagation, the speed and distance of cracks’ dynamic propagation and crack arrest. Subsequently, in Section 3, we show the newest numerical models and their advances in research into the formation of avalanches. Numerical simulation can accurately capture the dynamic crack propagation and stress changes within the snow layer. In Section 4, we discuss the existing controversies and contradictions. For instance, one is a different view of weak layer fracture stress patterns and critical length variation with slope angle. Another is proposing the idea of modelling the unity of the two processes of the formation and flow of avalanches. Finally, there is a summary and outlook. Through a concise description of the avalanche formation process, our eventual aim is to gain a clear understanding of the avalanche formation mechanism and provide guidance for future avalanche formation studies.

2. Dynamic Processes of Avalanche Formation

In this section, we provide an overview and discuss the physical processes and important variables at different stages of the formation of avalanches. The investigation of physical quantities includes different approaches, theoretical and experimental, and a few numerical results for comparison. Theoretical development has gone through the three vital periods of pure shear, anticrack and sub-Rayleigh to supershear transition. Among them, the pure shear model [19] was the first to apply the fracture mechanics theory to the study of the formation of avalanches. It is illustrated that the shear fracture failure of the weak layer is the main contributor to the slab avalanche. The anticrack model [20] suggests that in addition to the shear stress applied to the weak layer, the compressive stress exerted by the snow slab also contributes to the collapse. The subsequent research has gradually shown the formation of avalanches as a mixed-mode anticrack propagation [8,40,54,55]. Recently, Trottet et al. [37] proposed the existence of a crack propagation transition from sub-Rayleigh to supershear. It is demonstrated that there is also a transition from mixed mode to pure shear in the stress pattern. Experiments have also become one of the main approaches to avalanche formation research in the last two decades. Laboratory experiments have focused on the mechanical properties of snowpack fracture failure, whereas field experiments have concentrated on the fracture extension process at a small scale. Early experiments focused on the stability conditions of snowpack, including acoustic emission [43], the Rutschblock test [56] and the use of the snow micro-penetrometer (SMP) [57], which qualitatively evaluates avalanches from snowpack stability. Following the Compression Test (CT) [58] and Extended Column Test (ECT) [59,60], the processes of the onset of weak layer failure and crack propagation were investigated, respectively. At present, the Propagation Saw Test (PST) [61,62] is the predominant experimental approach. The overall process of the onset of failure and dynamic propagation of the weak layer until the slab’s tensile fracture can be better analysed by using PST. Theory and experiment have been developed together to validate and quantify several key parameters at different stages of the formation of avalanches. The numerical tools of the discrete element method (DEM) and material point method (MPM) also provide validation results accordingly. We discuss the separate research components in different stages and sort out the developmental timeline. This also contains the progression of skier-triggered avalanches.

2.1. Critical Crack Length

The release of dry snow slab avalanches involves the formation of local damage within a weak layer. Initial damage leads to cracks’ formation in the weak layer, and the stress concentration determines whether the cracks will continue to propagate and eventually trigger instability. The critical length is the minimum value for the continuous propagation of the crack, that is, the beginning of dynamic continuous propagation. Hence, critical length is a vital parameter to evaluate snowpack stability [63]. Theoretically, the first pioneer to apply fracture mechanics to the investigation of the formation of avalanches was McClung [19], firstly proposing a stronger snow slab over a weak layer. The Griffith-like slow shear deformation in the weak layer is the key to its failure and the subsequent avalanche release [19,64], as shown in Figure 2a. Shear bands are generated at stress concentrations in the weak layer, and a slow strain softening occurs at the tip of the shear band. Until a critical length is reached, the shear zone propagates rapidly, thus triggering the avalanche. Since then, the critical crack length has become one of the most important objects to understand the onset of failure [25,37]. Experimental observation of strain softening revealed that avalanche release may occur with or without additional loading when stresses are concentrated in the weak layer [65,66]. The shear model analyses the initial strain softening criterion for the weak layer and subsequently gives the following expression for the critical crack length based on the shear fracture theory [65].

$$a_c={\displaystyle \frac{D}{{\tau }_g-{\tau }_r}}\sqrt{{\displaystyle \frac{4G}{D(1-\nu )}}({\tau }_p-{\tau }_r)\delta }$$

(1)

where $G$ is the shear module, $\nu$ is the Poisson’s ratio, $D$ is the slab thickness, and ${\tau }_g=\rho gD\sin \psi$ is the shear stress due to the gravity. ${\tau }_p$ is the peak stress, and ${\tau }_r$ is residual stress. $\delta$ represents the displacement of the shear zone from peak stress to residual stress. It is necessary to specify a non-singular stress field at the tip of a shear zone requiring an end zone or plastic zone of length $\omega$. In this zone, stress concentration could lead to tensile fracture of the slab, thus evoking an avalanche. Based on the pure shear model of McClung [19], Chiaia et al. [67] assumed a pre-existing weak layer underlying the slab. The stress failure criterion and the energy criterion were introduced to predict avalanche triggering, and these two criteria were coupled together. This indicates that failure of the weak layer can only arise within a specific thickness range, which also refines the pure shear fracture model further.

The shear model states that the major factor in the formation of avalanches is that the weak layer shear stress exceeds the strength. However, weak layer collapses and the sudden subsidence of the slab can also occur in horizontal terrain and spread over long distances. Thus, pure shear theory does not fully explain the avalanche formation process. After a long period of shear fracture as the main traditional mechanism for slab release, Heierli et al. [20] proposed the anticrack nucleation model as another important trigger mechanism to induce avalanche release (Figure 2b). The anticrack model assumes a finite thickness of the weak layer, and failure is driven by a combination of gravity-induced shear and compression components. The weak layer experiences volume collapse during the failure process and is accompanied by a “whumpf” sound [20,68]. The pure shear model considers only type II cracks. However, unlike other materials, snow can fracture under compression due to its high porosity and deformability [42]. Hence, type I cracks also have an important effect. Also known as anticrack because of the opposite displacement direction, the anticrack model assumes that the thickness is reduced by the weak layer collapse, and it gives the expression of the critical crack length as follows [20].

$$a_c={\displaystyle \frac{4}{\pi \gamma }}{\displaystyle \frac{w_{f}E}{{\left(\rho gD\right)}^2}}$$

(2)

where $w_f$ is the specific fracture energy per unit of the fracture surface, $E$ is the elastic modulus of the snow slab, $\gamma$ is a constant with an approximate value of 1, $g$ is the acceleration of gravity, and $\rho$ is slab density. The anticrack model also suggests that skiers moving on flat or low-angle terrain may trigger remote avalanche on adjacent slopes, explaining the avalanche remote triggering phenomenon [21,26,69]. The proposed anticrack model provides a more profound theoretical basis for avalanches’ formation and release and dramatically improves our understanding of those processes. Since the PST experiment was designed, Sigrist and Schweizer [62] have only begun to measure specific values of critical crack length (Figure 3). It was also used to calculate the critical energy release rate. From the PST experiment, they measured that the critical length of rapid crack propagation was about 25 cm, and the specific value of the critical crack length was measured for the first time from the experiment.

Subsequently, Gauthier and Jamieson [70] obtained critical lengths in the range of 13–95 cm from more than 600 crack expansion experiments. In addition, the effects of slope angle and cutting direction (up and down) on the test results were evaluated in PST experiments, and a standard column geometry for testing was proposed [71]. Following researchers have successively considered the size of the critical crack length under different snowpack conditions. Based on anticrack [20], van Herwijnen et al. [72] measured the crack propagation deformation field in a snow layer under low-angle terrain. The collapse and deformation of the slab during crack propagation was also analysed using high-speed photography and particle tracking velocimetry (PTV). A critical length range between 16 and 44 cm was obtained. However, no shear fractures prior to collapse were detected, contradicting the previous understanding that fracture propagation was driven by shear fracture [19]. Subsequently, van Herwijnen et al. [73] proposed a method based on field experiments to determine the effective elastic modulus of the slab and the specific fracture energy of the weak layer. Furthermore, a critical crack length mean value of 28 cm was measured using a modified PST experiment. Schweizer et al. [40] also performed a field PST to determine the mechanical properties of the slab and the weak layer. It is shown that the crack propagation propensity is the result of a complex interaction of slab and weak layer properties, rather than being determined by a single mechanical property. The critical length measured at different times was between 20 and 50 cm and was compared with data from the SMP. Reuter et al. [63] obtained critical crack lengths in the range of 10–60 cm from SMP measurements, which are relatively consistent with actual measurements. The variation in critical crack length and specific fracture energy with different parameters is shown in

Table 1. The values of critical crack length and specific fracture energy with different parameters.

Literature	Methods	Slab Density (kg/m3)	Slab Elasticity Modulus (MPa)	Slab Thickness (cm)	Weak Layer Thickness (cm)	Slope Angle ($\mathit{\theta }$)	Critical Crack Length (cm)	Specific Fracture Energy (J/m2)
Sigrist and Schweizer [62]	PST (EXP)	187	7.5 ± 2.5	26	0.2	30	23 ± 2	0.07 ± 0.2
Gauthier and Jamieson [70]	PST (EXP)	134	1.5 ± 0.8	14	2–4	0	13	0.03
						30	29	-
						38	22	-
Chiaia et al. [67]	Theory	200	0.5–10	0.5	1	30–45	60–70	0.1–0.3 (mode II)
Heierli et al. [20]	Notch experiments	187	7.5 ± 2.5	26	Rigid WL	30	29 ± 5	0.1
van Herwijnen et al. [72]	PST	150–250	-	38–66	1–9	0–19	16–44	-
Reuter et al. [63]	SMP	50–400	16	10–200	-	30	10–60	0.07–2.9
van Herwijnen et al. [73]	PST	73–316	0.08–34	10–150	Surface hoar (39.6%)	0–50	2–89	0.08–2.7
Schweizer et al. [40]	PST	207–309	2.5–10	56–148	5–8	Field topography	20–50	0.5–1.4 (SMP)
Gaume et al. [25]	DEM	50–500	0.1–50	20–100	3–6	0	22	- (material strength)
						10	18
						20	12
						30	8
Gaume et al. [55]	MPM	180	2	20	2	0–50	0–60	-
Birkeland et al. [75]	Load-PST (EXP)	157–246 253–340 (loading)	0.14–2.5	12–40 40–60 (loading)	4	20–27	4.5–25.8	0.07–0.9
Bobillier et al. [26]	DEM	154	5.2	110	2	0	28	-
Trottet et al. [37]	MPM	250	5–30	50	12.5	0–50	(300–500) super crack	-

. Even though the anticrack model [20] is widely used to study the formation of avalanches, the model neglects the elastic mismatch between the snow slab and the weak layer. Anticrack points out a low dependence of the critical crack length on the slope angle, which is not in conformity with practical observations [25]. Accordingly, Gaume et al. [25] proposed a new expression for the critical crack length through a discrete element method (DEM) model. As shown in Equation (3),

$$a_c=\Lambda \left[{\displaystyle \frac{-{\tau }_g+\sqrt{{\tau }_g^2+2{\sigma }_n({\tau }_p-{\tau }_g)}}{{\sigma }_n}}\right]$$

(3)

where ${\sigma }_n$ is the normal stress, and $\Lambda$ is the length scale, representing the characteristic length of shear stress exponential attenuation near the crack tip. This model has explained the pivotal physical process of crack propagation in the weak layer, which is the complex mechanical behaviour of the weak layer and mixed-mode stress state in slab tension and bending caused by the weak layer’s collapse. Results were given that the critical crack length decreases with the slope angle. The strong dependence of the slope angle on the critical length was also revealed. The general real slope angle range for cracks’ extension is between 4 and 49° [74].

Then, Gaume et al. [36] also demonstrated the critical crack length as a function of the slope angle by a different numerical method (the finite element method, FEM). The different shear strengths were taken separately, and the results similarly decreased the critical crack length with increasing slope angles. This is consistent with the general experimental observation. By improving the DEM model, Bobillier et al. [26] simulated a critical crack length of 28 cm. Because of the fast response of the DEM particle-bonding system, it accelerates the rapid crack extension time in the weak layer. Thus, the simulation results are lower than the 32 cm as measured in the experiment. The previous PSTs were quasi-static experiments, i.e., a fixed snow slab layer thickness was chosen. Nevertheless, Birkeland et al. [75] investigated how loading affects snow’s mechanical properties in relation to crack propagation. This includes the changes immediately after the loading of disaggregated snow and in the following days. The outcomes display that increasing the snow load decreases the critical crack length and increases the crack propagation speed. The variation in critical crack length is essentially driven by an increase in the snowpack elastic modulus and weak layer fracture energy [75]. It also helps us to understand the critical state of crack expansion.

All previous studies on critical crack lengths were in sub-Rayleigh regimes, i.e., in a low-angle topography or shorter column length. Trottet et al. [37] showed the existence of a transition from sub-Rayleigh anticrack to supershear propagation (Figure 4). One of the conditions for transition to happen is that a super-critical crack length is reached. After reaching the super-critical crack length, the stress distribution near the crack tip undergoes a clear and sharp transformation [37,46,47]. At this juncture, the stress pattern undergoes a transition from mixed-mode anticrack to pure shear. Furthermore, the super-critical crack length demonstrates a notable correlation with the slope angle, as depicted in Figure 4. The super-critical length decreases with an increasing slope angle, which is in accordance with the findings under sub-Rayleigh. The expression for the super-critical crack length is given by the following equation [37]:

$$a_{sc}=\Lambda {\displaystyle \frac{{\tau }_p-{\tau }_g}{{\tau }_g-{\tau }_r^{\ast }}}$$

(4)

where ${\tau }_r^{\ast }={\tau }_0\cos \psi \tan {\varphi }^{\ast }$ is the effective residual stress in the shear zone and a function of the slab load and the effective friction angle ${\varphi }^{\ast }$. The supershear transition theory bridges the gap between small-scale PST and real avalanche experiments. This model reconciles the advantages of the pure shear model and the anticrack model and better describes the dynamics of the formation of avalanches.

2.2. Crack Propagation Speed

The crack begins to dynamically self-expand once it reaches a critical length. The crack propagation speed is one of the most important parameters for the dynamic propagation of cracks in the weak layer. The magnitude of the crack propagation speed also determines the timing and size of the avalanche. When the slab is strongly heterogeneous, a fracture of the slab may appear, at which time the crack propagation speed is reduced, or the crack arrest phenomenon occurs. Conversely, a higher propagation speed in the weak layer leads to the formation of large-scale avalanches. The crack propagation speed in the weak layer is mainly classified into two regimes, as shown in Figure 2, which are sub-Rayleigh and supershear propagation. The primary concern in the early days was the speed of crack expansion in the sub-Rayleigh regime.

Johnson et al. [43] were the first to measure the speed in horizontal terrain by using seismic sensors (Figure 5a). An artificial perturbation is introduced at the centre of the experiment in order to trigger a “whumpf” event. The time difference between the signals received by each geophone is measured and employed in the calculation of the propagation speed of the fracture. The cracks extended over a distance of more than 8 m with a measured speed of approximately 20 m/s. In an experiment conducted by van Herwijnen and Schweizer [76], the use of seismic sensors enabled the measurement of crack propagation velocities in the range of 38–46 m/s at a distance of over 60 m. As an alternative approach, the crack propagation velocity was derived by measuring the vertical displacement thresholds at the marked points in isolated beams [77]. This resulted in propagation velocities in the range of 17–26 m/s, which is also in agreement with the previous measurements. In small-scale PST experiments, a combination of high-speed video and particle tracking velocimetry has become the prevailing methodology for measuring crack extension speed [72]. The velocities were found to be within the range of 20 to 60 m/s, as determined by measuring the slab displacement field in low-angle terrain. The results show that the volume-conserving pure shear model is not applicable to the crack propagation in low-angle terrain, nor can it account for the remote triggering of crack propagation from horizontal terrain to slopes. Although this method addresses anticrack propagation in low-angle terrain, however, avalanches all occur in high-angle terrain, so this model also weakens the contribution of shear fracture.

In addition to the PST, the Extended Column Test (ECT) experiment was also used to measure the crack propagation speed [78] (Figure 5c). To ascertain the stability of the snow column, ECT employed a tapping technique. The crack propagation velocities obtained from the ECT experiments were found to be within the range of 20–30 m/s, thereby demonstrating that the fracture velocities of ECT are comparable to those measured by PST. Bair et al. [79] employed ECT and PST experiments to measure a crack extension speed in the range of 14–35 m/s, a result that is in close approximation to previous findings. Furthermore, he evaluated the impact of edge effects on crack propagation in snowpack stability tests. The outcomes illustrate that the percentage of tests exhibiting complete crack propagation declines with the column length. This is attributed to edge effects, which result in a greater collapse at the extremities of the beam than at its centre. Since then, PST experiments have also had to take edge effects into account. The variation in crack propagation speed with different properties is shown in

Table 2. Crack propagation speed as a function of different properties.

Type	Column Length (m)	Slab Thickness (WL) (cm)	Average Slab Density (kg/m−3)	Crack Propagation Speed (m/s)	WL Crystal Structure	Literature
whumpfs	12.7	40 (1)	190	20 ± 2	SH	Johnson et al. [43]
compression tests, Rutschblock and cantilever beam	0.3–2	42–94 (0.7–2.1)	171.25	17–26	SH, FC and DH	van Herwijnen and Jamieson [77]
ECT	0.9	32–67	143–316	20–30	SH, FC and DH	van Herwijnen and Birkeland [78]
	3–7	45–58 (6)	216–249	14–35	FC	Bair et al. [50]
PST	3–4	38–66 (1–9)	150–250	20–60	SH and FC	van Herwijnen et al. [72]
	2.3–3.3	0.23–0.74	138–149	20–30	SH	Bergfeld et al. [44]
	1–400	0.88–1.09 (1.5–21)	157–181	30–54	SH, PP and RG	Bergfeld et al. [45]
	9	23–109 (1–1.5)	110–360	32–40	SH,	Bergfeld et al. [29]
DEM	2	20–100 (1–6)	250	10–50	triangle structure	Gaume et al. [21]
	4.35	110 (2)	154	42	cohesive ballistic (SH)	Bobillier et al. [26]
	5–20	40 (2)	250	0.6–1.6cs	cohesive ballistic (SH)	Bobillier et al. [46]
MPM	25–140	50 (12.5)	250	0.4–1.6cs	SH	Trottet et al. [37]
Avalanche experiment	12–590	-	-	18–428	-	Hamre et al. [48]
	400	88 (21)	157	36	RG	Bergfeld et al. [45]
	6–54	-	-	11–250	-	Simenhois et al. [80]

. In order to reduce the influence of edge effects, Bergfeld et al. [44] extended the PST experiment to a distance of 2.3–3.3 m (Figure 5d). A high-speed camera and digital image correlation (DIC) were employed to obtain high-precision displacement and strain fields, which were then used to calculate the crack extension velocity and touchdown distance. The experiment allows for the observation of three distinct fracture modes: slab fracture, crack arrest and full propagation. The propagation velocities were also quantified in the range of 20 to 30 m/s. Subsequently, the velocity values for crack propagation were measured at multiple scales (1–400 m) in order to gain insights into the formation of avalanches [45]. From the PST experiment, Whumfp and real avalanche, respectively, they found average crack velocities between 36 ± 6 and 49 ± 5 m/s, and the velocities did not show a scale dependence. This experimental comparison incorporates a comprehensive spectrum of experimental conditions, representing a significant advancement in crack speed measurement. While the multi-scale propagation velocities are captured, the underlying reason for the discrepancy between the crack expansion velocity and the shear wave velocity in high-angle topography remains unclear. Furthermore, the morphology of the crack expansion is not adequately recognised. Afterwards, they further increased the column length to 9 m and measured a crack propagation speed of about 36 m/s [29] and proposed the self-sustained crack propagation (SSP) index. When the snow slab is softer and has a lower tensile strength or insufficient loading, the snow layer is not susceptible to crack propagation but rather to snow slab fracture or crack arrest, which reduces the propagation speed and the avalanche’s magnitude. When the SSP exceeds a specified threshold value, the crack is more prone to self-sustained dynamic propagation, which may result in the formation of large-scale avalanches [29].

The enhancement in computational capability has been accompanied by a progressive development in the range of methodologies for quantifying crack propagation speed through numerical simulation. The discrete element method, which is based on the Lagrangian particle tracking method, is capable of accurately capturing the velocity and displacement of each particle. This enables the calculation of the process of weak layer failure and dynamic crack propagation. Gaume et al. [21] pioneered the development of DEM^2D to assess the response of crack propagation velocity to the specific snowpack characteristics and slope angle. In particular, the crack propagation speed is observed to increase with an increase in the elastic modulus, thickness, density and slope angle of the snow slab, and conversely, to decrease in correlation with an increase in the strength of the weak layer. A relatively wide velocity range (10–50 m/s) was obtained by DEM^2D. Since then, DEM^3D has been employed for the investigation of crack propagation in the weak layer [26,81]. The weak layer is generated by the viscous ballistic deposition method, which is able to reflect the anisotropic characteristics of the weak layer. There are four ways defined to calculate crack propagation speed, which are the bond-breaking position, normal displacement thresholds, maximum stress and maximum normal slab acceleration [26]. The stabilised propagation speeds measured by the different methods are close to each other and are about 40 m/s. The variation in normalised crack propagation speed with slope angle is shown in Figure 6. $c_s^n$ is the normalised shear wave speed, which is calculated as follows:

$$c_s^n=\sqrt{{\displaystyle \frac{E/2(1+\nu )}{\rho }}}$$

(5)

where $\rho =250\hspace{0.17em}$ kg m⁻³ is the slab density; in order to calculate the elastic modulus, values were selected from the existing literature [82].

$$E=5.07\times {10}^9{({\displaystyle \frac{\rho }{{\rho }_{ice}}})}^{5.13}$$

(6)

${\rho }_{ice}=917$ kg m⁻³ is the ice density; we used a Poisson’s ratio $\nu =0.3$ [37,46,47]. As can be seen, in low-angle topography and shorter PST experiments, the crack propagation is in a sub-Rayleigh regime, with propagation speeds ranging from 0.1 to 0.5 shear wave speeds.

The crack propagation speed previously measured from small-scale PST experiments and ECT experiments were in the sub-Rayleigh regime. The propagation speed was significantly smaller than the shear wave speed [83]. However, Hamre et al. [48] obtained propagation speeds ranging from 18 to 428 m/s from real-scale avalanche data. This value is markedly higher than those previously reported by PSTs and also exceeds the snowpack shear wave velocity. This experiment was triggered by an artificial explosion. For this observation, photographs were taken from an observational video measuring the time between the explosion and the break-up of the snow blocks. A combination of in situ measurements, image scaling and Google Earth measurements was used to calculate the propagation speed. Most of the measured speed range was between 50 and 125 m/s, which exceeds the shear wave speed of snow. Despite the potential for inaccuracies in the measurement technique to impact the reliability of the estimated crack speed, the method remains a viable means of assessing the speed. However, this discrepancy in velocity has prompted further attention and generated considerable debate. Furthermore, this has prompted a re-evaluation of the mechanism underlying the formation of avalanches, with a particular focus on whether the observed phenomena can be attributed to mode I or mode II. To account for the effect of slope angle, Gaume et al. [84] proposed a unified model based on the material point method (MPM) to study avalanches’ release and flow at the slope scale. The simulation results indicate crack propagation speeds of up to 150 m/s along the slope, which is approximately three times the magnitude of the field experiments. This finding also suggests that crack propagation speed can exceed shear wave speed, and that avalanches are more probable with larger slopes [80].

Trottet et al. [37] importantly pointed out the presence of a transition from sub-Rayleigh anticrack to supershear crack propagation in the weak layers. This finding provides a rationale for the discrepancy in propagation speeds between horizontal and slope terrain at both small and real scales, representing a significant advancement in our comprehension of the formation of avalanches. Under terrain below the effective friction angle, crack propagation is in the sub-Rayleigh regime in the weak layer. The propagation speed is about 0.4 times the shear wave speed, and cross-slope propagation in real terrain follows this pattern. Nevertheless, as the angle continues to increase, the crack propagation pattern undergoes a transformation into supershear, as shown in Figure 7. The propagation speed of cracks is up to 1.6 times the shear wave speed. Certainly, there are two conditions for this transformation, the first being that the slope angle exceeds the effective internal friction angle of the snowpack. Secondly, the snow column is long enough to reach a super-critical crack length. The simplification of the snowpack to a finite-thickness Timoshenko beam model has also helped us to recognise this translation process [34]. The compaction of the weak layer may serve to reduce the anticrack speed, while the weak layer’s elasticity can lead to a higher speed. Additionally, Bobillier et al. [46] employed DEM to quantify the impact of snowpack properties on dynamic cracking in sloping terrain. This extends the numerical PST model to 20 m, and the results exhibited a propagation speed for two different regimes. In flat terrain, the propagation of weak layer fractures is governed by compressive stress, with the speed of crack growth contingent upon the elastic properties of the overlay or weak layer and the strength of the weak layer itself. On the steep slope, the shear stress of the weak layer exceeds the compressive stress, which also indicates the existence of the fracture mechanism of supershear expansion. Although there has been a further breakthrough in cracks’ dynamic propagation, there are obvious differences between small-scale PST experiments and real field observations, and the theoretical unification between the two still needs to integrate many factors.

2.3. Crack Arrest and Propagation Distance

When the crack is dynamically propagated in the weak layer, the snow slab is fractured by gravity stretching the slab down the slope, where the crack propagation speed decreases or the propagation arrests. Subsequently, the released snow volume slides down the slope to form an avalanche [6]. Thus the snow slab fracture is the foundation of the formation of avalanches, and the magnitude of the propagation distance determines the release size. There is a relationship between crack length and crack arrest with test conditions and slope [61]. The capacity of a slab to convey damage from one place to another through a stopping condition is expressed as the sustainability of the fracture. Early PST experiments were mostly full propagation due to the sufficiently high density and strength of the snow slab layer and its short length. It is uncommon for a slab to fracture due to the crack’s full propagation. However, in recent years, after choosing a slab with low density and strength, snow slab fracture and crack arrest have also appeared in the PST experiments. In addition, the slab fracture and crack arrest processes also occurred after increasing the length of the classical PST column [29,44].

The first measurement of propagation distance derived from a numerical PST was conducted by Gaume et al. [21]. A three-layer model was constructed using DEM in order to recreate the process of cracking propagation that was observed in the field PST. The impact of the varying snowpack layer properties on the extended distance response to each parameter was quantified. Among the variables, those pertaining to the tensile strength, elastic modulus and density of the snow slab layer exert a significant influence on the extension distance, as depicted in Figure 8a. The results also show that the propagation distance increases with increasing slab tensile strength and density and decreases with increasing elastic modulus. This illustrates the fact that harder slabs propagate further, and softer slabs are more prone to fracture [21,50], and therefore more subject to crack arrest. The propagate distance is also quantified based on theoretical models and experiments. Birkeland et al. [85] performed repeated PST tests on a weak layer buried in surface hoar, focusing on fracture arrest conditions. As stated by Reuter et al. [63], the characterisation of unstable snowpack conditions requires the presence of both high damage initiation and high crack propagation propensity. This unique condition allows for the observation of dynamic crack propagation in the weak layer. Benedetti et al. [86] developed an analytical model of the PST based on the Euler–Bernoulli beam theory. This was conducted in order to calculate the critical crack length and the crack propagation distance as a function of the snowpack characteristics and the beam geometry. The model reproduces the trend observed in field experiments of increasing distance with increasing slab density. In the case of a low-density slab, fracture occurs prior to the critical crack length being reached. For moderate density values, snow slab fracture occurs after weak layer crack propagation has begun. However, the high slab density results in a complete extension in the weak layer, whereby the snow slab does not break and the crack propagation continues to the end, which is also consistent with the findings of Gaume et al. [21]. Figure 8b shows the variation in dimensionless propagation distance for different slab densities. The data indicate that lower densities are more prone to fracture and crack arrest, while higher densities tend to be more fully propagated. The fracture of the snow slab also reduces the propagation speed of cracks in the weak layer [44,83]. At the location of the fracture in the slab, the propagation speed decreases significantly. Nevertheless, a snow slab fracture does not necessarily prevent crack propagation and en echelon fracture types can emerge [37,87]. Accordingly, the establishment of a self-sustained propagation (SSP) index, which considers the relationship between crack propagation speed and critical crack length, is an effective method for describing the fracture and crack arrest of the snow slab [29]. The expression for the tendency of snowpack to support self-sustaining crack propagation is as follows:

$$SSP={\displaystyle \frac{v_c^2}{g\hspace{0.33em}{a}_c}}$$

(7)

When the self-sustained propagation index is high, the crack propagation is more inclined to be fully extended. The SSP index represents the inaugural attempt to estimate the propensity of snowpack propagation. While the model may assist in comprehending the expansion tendency, it is important to note that the dataset utilised exclusively encompasses experiments conducted on flat terrain. This method does not take into account the fact that it is performed on slopes with different angles, so its application in avalanche release modelling is not sufficiently clear. In addition, this index formula lacks elucidative plausibility, because the propagation distance is correlated with covariates such as slab elastic modulus, strength and density [21].

In contrast, the critical crack length and propagation distance are two independent variables that are used to determine the initiation and eventual cessation of cracks. However, these variables lack the necessary relevance as they are not inherent properties of the snowpack. While the propagation speed can be indicative of the extension distance, there is no direct correlation between the two. Accordingly, it is possible for cracks to be propagated a long distance at lower speed. Recently, Meloche et al. [88] have considered two different types of slab, purely elastic and plastic, and proposed a scaling law for the relationship between the crack arrest distance and weak layer strength variability and the dimensionless number associated with slab tensile fractures. This approach can be effectively employed to quantify the interplay between weak layer heterogeneity and snow slab layer tensile fractures. At present, propagation distance is obtained from PST experiments, and it is challenging to comprehend the full extent of these distances in real scale. In the future, it may be appropriate to consider the use of numerical modelling to capture the response of slab fracture, crack arrest, and other relevant features under different snow layer properties and slope angles.

2.4. Energy Release Rate and Specific Fracture Energy

The formation of dry snow slab avalanches is a result of a series of fracture processes in the snow layer [6]. In a previous section, the stages of the formation of avalanches were quantified from a mechanical perspective, and the critical crack length was used to express the tendency for crack propagation. The weak layer’s failure and snow slab layer’s fracture entail the release and transformation of energy, making it crucial to quantify the avalanche fracture process from an energy perspective. Similarly, the specific fracture energy of the weak layer is a significant factor in determining the stability of the snowpack. The specific fracture energy is an intrinsic property of the snowpack, defined as the energy required to fracture a unit area of the weak layer. The energy required for the fracturing of the weak layer is provided by the upper snow slab layer. The potential energy stored in the slab is transformed into kinetic energy, which drives the propagation of cracks at the tip. The variation in the energy release rate with column length for different types of slab configuration is shown in Figure 9a. It is only when the energy release rate exceeds the specific fracture energy that crack propagation can proceed dynamically [29]. The specific fracture energy is also a crucial indicator of the rapid crack propagation during the release of an avalanche [62]. Considering different fracture modes separately, the specific fracture energy is calculated as in the following equation [29].

$$\begin{array}{l}{w}_f=G_{\mathrm{I}}+G_{\mathrm{II}}\\ G_{\mathrm{I}}={\displaystyle \frac{D_{WL}}{2E_{WL}}}{\sigma }_{WL}{(r=r_c)}^2\\ G_{\mathrm{II}}={\displaystyle \frac{D_{WL}({\nu }_{WL}-1)}{2E_{WL}}}{\tau }_{WL}{(r=r_c)}^2\end{array}$$

(8)

where $G_{\mathrm{I}}$ and $G_{\mathrm{II}}$ are the contributions from mode I and mode II, respectively, and ${\sigma }_{WL},\hspace{0.33em}{\tau }_{WL}$ are the compressive and shear stress in the weak layer at the crack tip. The first experimental measurement of the specific fracture energy of a weak layer was conducted by Sigrist and Schweizer [62].

The critical crack length was determined through an independently developed PST experiment [61]. Subsequently, the critical energy release rate corresponding to the critical cut length was calculated using finite element modelling. The measured average critical energy release rate was approximately 0.07 J m⁻². This is the crucial breakthrough in which the specific fracture energy has been derived from a field test. It facilitates our comprehension of the fracture and crack propagation of the weak layer from an energetic perspective. Subsequently, Schweizer et al. [89] performed 150 PST experiments using the same method. The specific fracture energy level was determined to be 1 J m⁻², which is markedly higher than the previously recorded values. The discrepancy in the results measured between the two is mainly due to the lower effective modulus used by the latter [73]. Using signals from a snow micro-penetrometer (SMP), Reuter et al. [63] estimated a weak layer specific fracture energy in the range of 0.5–2 J m⁻². In light of the considerable difference in the experimentally determined specific fracture energy, van Herwijnen et al. [73] conducted a modified PST to assess the specific fracture energy of the weak layer and the effective elastic modulus of the slab. Particle tracking velocimetry was employed to analyse the displacement of markers within the snow slab and to derive the mechanical properties, where the effective elastic modulus of the slab ranged from 0.08 to 34 MPa, increasing with the average slab density. The specific fracture energy of the weak layer ranged from 0.08 to 2.7 J m⁻² and also increased with the increase in the cover layer density (Figure 9b). However, the analysis did not consider the impact of nonlinear deformation on energy dissipation, resulting in a measured weak layer that is biased higher than the fracture energy. In recent years, Bergfeld et al. [29] also measured a weak layer specific fracture energy in the range of 0.1–1.5 J m⁻² by an extended PST experiment (9 m). The weak layer specific fracture energy similarly increases with increasing density (Figure 9b). Furthermore, he evaluated two distinct dissipation mechanisms: dynamic fracture and compaction dissipation. In particular, the dynamic fracture energy is the energy required to destroy the weak layer in the region of the crack front. Compaction dissipation is the additional energy required for further fragmentation and compaction of the weak layer in the region behind the crack [29]. There is a different energy mechanism that also provides a deeper perception. Nevertheless, current energy release and dissipation models do not take into account other forms of energy transformation, such as the internal energy of the snowpack layer. The energy increases the sliding friction between the slab layer and the weak layer, in addition to providing the fracture and volume deformation of the weak layer. In the future, energy modelling is also more appropriate to consider a wider range of means of energy transfer. We consider avalanches initiated by skiers, as shown in File S1.

3. Numerical Modelling

In the last two decades, numerical simulation has emerged as the most recent and effective method for studying the formation process of avalanches. Although numerical modelling has been applied to avalanche research for a relatively short period of time, it has provided useful insights into their formation mechanism. The utilisation of numerical tools to investigate the formation of avalanches offers a number of advantages. Primarily, it enables the overcoming of the temporal and spatial limitations typically associated with field experiments. Additionally, it facilitates the dynamic detection of contact force and fractures within the snowpack from a microscopic perspective. These internal stress states are difficult to obtain from experiments. The essential numerical tools include the finite element method (FEM) [7], the SNOWPACK model [22], the discrete element method (DEM) [21] and the material point method (MPM) [36], etc. The following sections briefly describe the application of each numerical tool to the formation of avalanches. Furthermore, it considers the future employment of numerical modelling.

3.1. FEM and SNOWPACK

The finite element method (FEM) has a long history in the field of mechanics, where it is a standard tool for analysing loading-deformation experiments. The FEM was initially employed in the investigation of skier-induced avalanches [7,90], whereby it was utilised to ascertain the stress distribution within a snowpack when subjected to skier loading. The slab displacement deformation field can also be modelled by FEM, which is commonly used to study crack propagation and energy release. With the proposed anticrack model [20], the energy release rate of the weak layer has been suggested as another important factor in the formation of avalanches. Bair et al. [79] used a FEM model to simulate the strain energy release rate at a fixed crack length. The simulation results point out that the strain energy release rate G decreases with the test length when the crack length is r. The boundary effects in crack propagation experiments are also regarded. Furthermore, a functional relationship exists between mechanical energy and both slope angle and crack length [73]. The FEM results exhibit a positive correlation between mechanical energy and increasing slope angle, as well as a negative correlation between mechanical energy and an increasing dimensionless crack length. This also provides another perspective to account for the energy changes in the formation of avalanches. Hagenmuller et al. [91] used FEM to simulate the structural failure of the snowpack and variation in snow tensile strength from a microscopic viewpoint. The microstructures experienced a mixture of tensile and compressive stress under macroscopic force, and plastic yielding was produced by the non-local failure of the modelled ice-crystal bonds either before or after the peak stress was reached. It also explains damage, fracture nucleation and variations in strength. Besides this, the damage parameters of the weak layer are also an important influence on failure [92]. Thus, FEM can also be used to simulate the damage to the weak layer under cyclic loading. In particular, the dynamic stress stretching induced by vibration sprouts against weak layer edge rupture.

The pre-existing cracks in the weak layer exert a substantial influence on the propagation and the stress concentration in the underlying weak layer. Gaume et al. [36] employed the FEM method to determine the maximum shear stress and slab bending. It is illustrated that the maximum shear stress increases with an increasing crack length, slab thickness, density, weak layer elastic modulus and slope angle. It decreases with the slab elastic modulus. The model allows remote triggering from flat (or low-angle) terrain. The critical crack length decreases with an increasing slope angle, which is also in agreement with the results of Gaume et al. [25]. In a recent study, Bergfeld et al. [45] employed the FEM to ascertain the propagation velocity of purely elastic waves. The resulting value for the crack propagation speed is 0.4 times the bending wave speed. The FEM can be used to study stress concentration and crack propagation in snowpack layers. However, the finite element method is based on a strictly continuous medium theory, whereas the snowpack layer is composed of bulk snow particles. The assumption of a homogeneous snowpack, which fails to account for its inherent heterogeneity, inevitably introduces errors in the study of avalanches’ formation mechanism.

The concept of SNOWPACK was first proposed by Lehning et al. [22], and it is a model based on a Lagrangian finite element implementation that solves the stationary heat transfer and settling equations. While this method cannot be employed directly to study the formation of avalanches, it can be effectively utilised to assess the stability of snow slopes, thereby providing strong support for avalanche warnings. The SNOWPACK model includes phase transitions and the transport of water vapour and liquid water. Emphasis is placed on the metamorphism of snow and its relation to mechanical properties such as thermal conductivity and viscosity [22,23,93,94]. Bartelt and Lehning [95] present the SNOWPACK model more systematically. The interaction of microstructure, temperature and sedimentation introduced in the SNOWPACK model [23,94] can be well used to assess the stable structure of the snowpack and individual physical processes. This model can also be used to predict the probability of the onset of avalanches from a snowpack perspective. SNOWPACK is driven by data from automatic weather stations. Meteorological stations measure wind, air temperature, relative humidity, snow depth, surface temperature and ground (soil) temperature. The stability index is then calculated based on the modelled snow layer, i.e., the modelled layer properties. A number of stability indices have been incorporated into SNOWPACK. In particular, the natural stability index SN38 and the skier stability index SK38 [96,97] are frequently employed for the evaluation of snowpack stability. Furthermore, the SNOWPACK model has been employed to evaluate the critical crack length of the weak layer [25]. On the other hand, Richter et al. [93] verified the experimentally obtained critical crack length using the SNOWPACK model (Figure 10).

The temporal evolution of the vertical profile of the critical crack length demonstrates an overall increasing trend in critical crack length for each layer, as snowfall leads to an increase in snow depth. There is also research on avalanches’ formation process from the standpoint of weak layer failure. Although SNOWPACK does not provide direct insights into the processes governing the formation of avalanches, it is a valuable tool for assessing the snowpack’s stability. Furthermore, it can be effectively employed in the planning of crack onset and avalanche warning services.

3.2. DEM and MPM

The primary numerical methods used to study the formation of avalanches in recent years have been the discrete element method (DEM) and the material point method (MPM). The first to apply the DEM to the investigation of crack propagation in the weak layer was Gaume et al. [21]. A two-dimensional DEM model was employed to develop a typical three-layer structure of the snowpack. Where the overlying layer is a uniformly distributed snow slab, the mechanical properties of the slab are homogeneous. The intermediate weak layer is constituted by a triangular granular structure, which exhibits high porosity and isotropic characteristics. The underlying substrate layer is rigid and serves to support the entire snowpack. The crack propagation speed and the propagation distance for a number of different snowpack properties were derived from DEM simulations. The results demonstrate that the speed of crack propagation is directly proportional to the elastic modulus and thickness of the slab. The propagation distance is inversely proportional to the slab elastic modulus, density and weak layer thickness (Figure 8b), and it is directly proportional to the slab tensile strength, thickness, weak layer strength and slope angle. Subsequently, Gaume et al. [25] applied this DEM model to investigate the response of the critical crack length to the snowpack properties. The model indicated a reduction in critical crack length with increasing slope angle (Figure 3), thereby demonstrating the dependence of critical crack length on slope angle. This is also different from the results of the anticrack model [20]. In comparison, the anticrack model assumes a weak layer that is completely rigid and of finite thickness. Furthermore, the damage criterion is independent of the slope angle, thereby ignoring the elastic mismatch between the snow slab and the weak layer. The DEM model also elucidates that the initial trigger damage is more prone to induce the release of avalanches in high-angle terrain, which is also corroborated by experiments [75].

In addition to the analysis of the dynamic crack propagation in the weak layer, Mede et al. [8] employed a DEM model to calculate the initial failure of the snow layer in a mixed loading. The damage patterns of the snow samples were observed by subjecting the samples to normal and tangential stress. When subjected to a stress level exceeding the threshold for normal stress, the material undergoes a structural collapse, resulting in the formation of a snow sample comprising particles that lack cohesion. From a microscopic point of view, the snow deformation is governed by the two competing processes of fragmentation and sintering. By modelling the snowpack layer as a discrete particle-bonding model, the snow deformation can be studied in terms of the breaking and healing of the bonds. The mixed-model loading state has a high strain rate associated with typical avalanche releases. This also validates the results obtained by the anticrack model [20]. The simulation provides a new way to develop the snow damage mechanism under mixed loading. Similarly, Mede et al. [98] employed the same methodology and observed shear-induced macroscopic collapse above a critical level of normal stress. When the value of the applied normal stress reaches a sufficient magnitude, the load-bearing part of the snow sample microstructure will eventually fail. In addition to the overall failure of the snowpack, the potential for weak layer failure has also been explored through a mixed model [99]. Likewise, the effects of mixed loading and rapid sintering on the destruction of weak layers were investigated using DEM. Accounting for the internal sintering mechanism of the snowpack, the number of fracture bonds and the strength of the weak layer at the failure place decreased significantly and independently of the loading rate as the loading angle increased. At a low loading angle and low loading rate, the weak layer strength increases dramatically, consistent with the characteristics of natural avalanches. For the shear-dominated loading mode, the effect of the loading rate on the damage behaviour is negligible. When loading slowly, the weak layer strength gradually increases due to the predominant role of the sintering mechanism [99]. Also in terms of the initial failure of snow layers, Bobillier et al. [9] analysed the mechanical behaviour of the snow slab and weak layer in the light of particle adhesion bonds. He employed the ballistic deposition method to generate a porous weak layer and dense snow slab. The objective of this study was to investigate the breaking bond changes and stress–strain relationships in the snow layer. Then, based on the same particle generation method, he investigated the weak layer’s crack propagation propensity [26]. The speed of crack propagation and the normal acceleration of the slab were measured by tracing the location of the crack tip and the displacement field of the slab. However, the effects of different snowpack factors and slope angles on crack propagation were not included. On the other hand, Bergfeld et al. [45] measured crack propagation speeds within 40 m/s using the DEM.

Since the mechanism of crack propagation’s transformation from sub-Rayleigh to supershear was discovered [37], Bobillier et al. [46,47] have also used DEM to capture this transformation process (Figure 11). From the numerical PST, it was found that the supershear propagation of crack occurs only when the column is long enough to go over the super-critical length, and the slope angle is higher than the effective friction angle. In this regime, shear stress become the dominant force, resulting in a crack propagation speed that exceeds the shear wave speed. This approach allows us to reconceptualise cracks’ dynamic propagation from a DEM perspective. The DEM enables the precise tracking of the particle’s movement within the snow layer. Additionally, the formation and rupture of the bonds between particles can be observed, providing insight into the fracture failure and sintering processes. Therefore, DEM is a mature and well-established method for assessing initial failure, as well as for investigating dynamic crack propagation and slab tensile fracture. This method can be well used to study the mechanism of the formation of avalanches.

The material point method (MPM) is a continuous hybrid Eulerian–Lagrangian numerical method that is well suited to the study of processes involving large deformations and fractures. The application of the MPM to the dynamics of crack propagation is relatively recent. Gaume et al. [36] for the first time investigated the weak layer’s crack propagation properties using the MPM and demonstrated that weak layer failure is a mixed-mode anticrack process. A crack develops in the weak layer and begins propagation. The weak layer then fails, and the snow slab layer is bent and settles internally. Finally, the progressive loss of slab support leads to the formation and release of an avalanche. The model applies the theory of elastic–plasticity and large strains in porous cohesive materials, which allows it to reproduce the initiation and dynamic expansion of fractures observed in snow fracture experiments [36]. Assuming that an initial crack of length r is pre-existing in the weak layer, the MPM can be used to determine the crack propagation speed [45]. The crack length is tracked by defining the crack tip as the location of the furthest plasticising particle, and the time evolution of the crack tip location is applied to directly obtain the crack speed c. The variation in crack speed with angle is shown in Figure 6. This also validates the results of the field PST experiment.

Trottet et al. [37] used MPM studies to show the presence of a transition between sub-Rayleigh to supershear crack propagation. This transition follows the Burridge–Andrews mechanism, whereby the supershear sub-crack nucleates before the leading edge of the main crack and eventually propagates faster than the shear wave speed. This represents the most current understanding of the formation of avalanches. A supershear propagation zone exists even when the shear–normal stress ratio is lower than the static friction coefficient due to a loss of frictional resistance during collapse. This transition occurs when the supershear critical length (Figure 4) and slope angle are achieved. The supershear contrasts with the much lower sub-Rayleigh values measured in the small-scale snow fracture experiment (PST). Furthermore, future research will seek to elucidate the distinction between small-scale experiment and large-scale observation, providing more compelling evidence to substantiate this distinction and establishing a clear link between the two theories. A Depth-Averaged Material Point Method (DAMPM) has also recently been used to evaluate the release of dry slab avalanches [100]. By combining MPM with the classical shallow water hypothesis, the large-deformation elastoplastic modelling of landslides is carried out in a computationally efficient way, to quantify crack propagation on slopes and the complex interaction between weak layer dynamic failure and slab fracture, thus assessing the shape and size of avalanche release areas in different terrains. It should be noted that the limitations of MPM include mesh sensitivity and non-local effects, as well as the necessity for more cumbersome contact modelling.

Above all, numerical simulation is an effective tool to investigate the formation of avalanches, especially the DEM and MPM. Even though numerical simulation has been used to analyse avalanche formation processes for a relatively short period of time, it has a large potential. The simulation work also complements experiments and facilitates the variables that are difficult to measure experimentally. In light of uncontrollable and dangerous avalanches, numerical simulation will also become a prevailing approach to investigate avalanches in the future.

4. Discussion

The comprehension of the processes underlying the formation of avalanches has undergone a period of evolution, culminating in the emergence of a relatively mature theoretical framework. However, there is still some controversy over the different stages of the formation of avalanches. For example, whether the stress pattern of weak layer failure is compressive or shear-dominated in the sub-Rayleigh regime, and whether it is a pure shear mode [37] or a mixed shear-dominated mode [47] after reaching supershear. In addition, there is no consistent agreement on the variation in the critical crack length with the slope angle. Ultimately, further investigation is required to establish a unified model of avalanches’ formation and flow. In this section, we discuss current controversies and present problems to provide guidance and assistance for further research on the formation of avalanches.

4.1. The Stress Pattern of Avalanche Formation

The stress analysis of the formation of avalanches suggests that avalanche is a mixed-mode fracture process, but exactly which one dominates is still controversial. Since the proposal of the shear fracture model [19], it has long been assumed that the formation of avalanches is the result of pure shear fracture [65,101,102,103,104], and that failure onsets when the shear stress in the weak layer exceeds the strength. Nevertheless, this model is unable to account for the phenomenon of “whumpfs”, whereby in low-angle terrain, the weak layer also fails by fracture and crack propagation along the weak layer. The anticrack model [20] was introduced to address this issue well. It considers the effect of normal stress, where the weak layer can collapse. This emphasises the crucial role of compressive stress. Furthermore, it provides an explanation of the phenomenon of remote triggering. Later models have also been based on mixed fracture patterns [10,105,106], suggesting that compressive and shear stress have an equally important function. It seems that compressive and shear stresses are already clear [83]. However, the anticrack model shows that the bending waves in the snow slab provide energy for crack propagation in the weak layer. Furthermore, the magnitude of the bending wave velocity is lower than the Rayleigh wave velocity, which is not in accordance with the results observed in real experiments [48]. Hence, the anticrack model cannot explain the phenomenon of a crack propagation speed higher than the shear wave speed. The latest supershear modelling [37] indicates that the transition from sub-Rayleigh to supershear occurs when the slope angle is sufficiently high and the extension distance is sufficiently long. At this point, the stress shifts from a mixed anticrack mode to a purely shear one. Nevertheless, Bobillier et al. [46,47] found that the stress at the contact interface was dominated by mixed-mode anticrack in the sub-Rayleigh regime, as detected by DEM. In contrast, the stress pattern becomes shear-dominated in the supershear regime, with compression and shear stresses existing together. It remains unclear whether the transformation to the supershear form is purely shear [35] or shear-dominated shear–compression coexistence [46,47]. Subsequently, additional numerical evidence is required to corroborate the interface stress pattern. In the future, numerical modelling should be the primary approach, because field PST experiments cannot be used to measure the variation in internal contact forces. This provides a more rational explanation of the stress patterns that occur during the formation of avalanches. Furthermore, the aim is to quantify the influence of snowpack parameters and terrain angle on the stress pattern.

4.2. The Response of Critical Crack Length to Slope Angle

Since McClung [19] proposed to study the formation of avalanches from the perspective of fracture mechanics, the critical crack length has been one of the quantities of greatest interest to researchers. However, the variation in critical crack length with slope angle has not been consistently concluded. McClung [19] derived an expression for the critical crack length from a purely elastic standpoint. Nevertheless, the variation in critical crack length with slope angle was not considered earlier. Van Herwijnen et al. [72] employed high-speed photography and particle tracking velocimetry to analyse the deformation of a slab layer in low-angle terrain during crack propagation. Yet no pre-collapse shear cracks were identified, which challenges the previous hypothesis that fracture propagation is driven by volume-holding shear cracks. Subsequently, the anticrack model of Heierli et al. [20] suggests that the slope angle essentially does not affect the critical length, or that the slope angle changes the critical length slightly, which is inconsistent with practical observations [25]. The model’s fundamental assumption is that the failure behaviour of the weak layer is independent of the slope angle. Additionally, it ignores the elastic mismatch between the snow slab and the weak layer, as well as the tensile and bending effects of the snow slab layer. Gaume et al. [25] have computed the variation in critical crack length by DEM, which shows that the critical length decreases with an increasing slope angle. A comparable outcome was achieved for our DEM model, as illustrated in Figure 3. The previous experimental and numerical PSTs almost investigated crack propagation in the upslope direction. Recently, Adam et al. [42], from the downslope direction, argued that the critical length increases with an increasing slope angle, where the upslope and the downslope direction are not symmetrical, and the result is the exact opposite of that previously observed [70]. In light of the ongoing debate surrounding the critical length, we propose that further extended PST experiments and numerical modelling are now required, simultaneously to verify the upslope and downslope directions, as well as the variation in the critical length with angle. In addition, the extended PST model is also able to capture the relationship of supercritical crack length with snowpack parameters and angle. Furthermore, the precise crack length can be employed to ascertain the energy release rate and specific fracture energy.

4.3. Towards a Unified Model of Avalanche Formation and Avalanche Flow

The scientific issues of avalanches mainly include avalanche formation and avalanche flow. It is imperative to integrate these two processes in order to develop a unified avalanche model. However, the gap between avalanche formation and flow in terms of theoretical approaches and experiments brings challenges to the alignment of the two. Avalanche real field experiments allow for the complete process from skier triggering to rapid crack propagation and slab tensile fracture and finally release and flow down the slope. A real field experiment can be well used to detect avalanche flow, including changes in flow pressure and speed. Nevertheless, as a field experiment is primarily based on remote camera observations, the observation is susceptible to scale and resolution. Thus, the real experiment does not capture the weak layer failure and crack propagation process during the formation of avalanches. The accuracy of the observations through slab fractures had a significant impact on the results, rendering the real experiments used to study the formation of avalanches inherently flawed. It is not possible to use small-scale experiments to assess avalanche flow, despite their suitability for crack propagation. Therefore, an experimental approach to associate the formation of avalanches with flow in a unified way is more difficult. In terms of numerical simulation, there is currently no unified model that can incorporate both the dynamic propagation of avalanche formation and flow processes. The application of numerical methods based on DEM and MPM allows for the effective simulation of the dynamic expansion process associated with the formation of small-scale PSTs. Similarly, the flow dynamics of large-scale avalanches can also be implemented. Accordingly, the deployment of numerical simulations is anticipated to facilitate the integration of avalanche formation and flow. The movement of an avalanche along the slope can be described as a high-Reynolds-number gravitational potential flow. A dry snow slab avalanche transforms into a powder avalanche moving up to 100 m in height and at speeds of up to 100 m/s, one of the most dangerous avalanche types [51,107]. It is also essential to consider wind field, surface types [11,108,109] and erosion and entrainment processes [110] during flow. In the future it is expected to propose a unified model for both processes through numerical simulation. However, both numerical methods have their respective drawbacks. For example, DEM requires greater computational resources to be deployed for small to large scale applications due to full-precision granular solving. The limitations of MPM include its sensitivity to mesh parameters, the necessity of accounting for non-local effects and the increased complexity of contact modelling. It is therefore necessary to reconcile the advantages and disadvantages of the various methods and to propose a more sophisticated unification model for avalanches. This will facilitate a comprehensive understanding of the avalanche dynamics process.

5. Conclusions and Outlook

The dry snow slab avalanche is the most common and dangerous type of avalanche, occurring in snow-covered highland mountains. The formation of a dry slab avalanche is the result of a series of fracture processes, which include four basic processes: the initial failure of the snow layer, the onset of crack propagation in the weak layer, the crack’s dynamic expansion, and the tensile fracture of the snow slab. Once the slab is released, it begins to slide down the slope and thereafter form an avalanche. Among the four processes of the formation of avalanches, the onset of crack propagation and dynamic propagation in the weak layer become the crucial ones for avalanches’ release. The different stages of the formation of avalanches have been studied through experiment and numerical stimulation, as well as the establishment of corresponding theoretical models. Nevertheless, there is no work that summarises the development of avalanche formation mechanisms in a more comprehensive way, which hinders researchers and enthusiasts from accurately understanding and working on the formation of avalanches. Therefore, we review the current situation of the formation of avalanches and the recent latest theories, as well as discussing current controversial perspectives and future research directions. The objective of our work is to provide guidance to avalanche researchers in the development of future research plans.

In this work, we are concerned with the quantities investigated in each of the four stages of the formation of avalanches. The primary components are the critical length at which crack propagation begins and the speed during the crack’s dynamic propagation. The energy release rate and the weak layer specific fracture energy at the onset of the expansion period also provide another viewpoint to recognise the process of fracture initiation. Subsequently, we elaborate on crack propagation distance and crack arrest, which determine the magnitude of the avalanche. Furthermore, numerical modelling has been employed to investigate the stress changes at the contact interface. We have sorted out these physical quantities of most interest in the formation of avalanches from different standpoints: theory, experiment and numerical simulation.

The shear model [19] suggests that the weak layer shear stress exceeds the peak strength before crack propagation begins. At this juncture, the crack reaches a critical length. Whereas, the anticrack model [20] states that weak layer failure is driven by a combination of gravity-induced shear and compression components. However, the critical crack length is considered to be independent of the slope angle, as elastic mismatch is not considered. After the PST experiment was designed [62], the specific value of the critical crack length was only measured in the field. Thereafter, the critical length has been measured for different snowpack conditions and slope angles [63,70,73]. The critical crack lengths determined through PST experiments exhibited a range of 10–90 cm (Table 1). Gaume et al. [25] employed the DEM to measure the decrease in critical crack length with an increasing slope angle and subsequently verified the results through experimentation and FEM [36]. However, Adam et al. [42], from the downslope direction, suggested that the critical length increases with an increasing slope angle. In the light of the current debate about the critical length, we concluded that there is a need for more extended PST experiments and numerical modelling at present. Meanwhile, it is able to capture the variation in super-critical crack length (3–5 m) [37]. The crack propagation speed is the quantity of most concern in the dynamic propagation stage, and early crack propagation speeds in the range of 10–50 m/s were obtained from acoustic emission and ECT and PST. This is due to the fact that the crack propagation is in a sub-Rayleigh regime under the mixed anticrack mode. Since then, numerical simulations have derived propagation speeds for different snowpack properties and slope angles (Figure 6). Nevertheless, real scale observations of propagation speeds ranging from 18–428 m/s [48] clearly exceed the shear wave speeds of the snow slab, indicating the transition to supershear in crack propagation [37]. The following step requires numerical PST experiments and a quantification of the effects of snowpack properties and slope angle on the transition mechanism. The propagation distance and crack arrest determine the magnitude of the formation of avalanches and are quantities that have received more attention in recent years. The propagation distance is observed to increase in accordance with the tensile strength and density of the snow slab. This suggests that the harder the slab, the greater the distance it is able to expand, while a softer slab is more susceptible to fracture [21,50]. Therefore, associating the crack propagation speed with the critical crack length to establish a self-sustained propagation index is also an effective way to describe slab fracture and crack arrest [29]. In the future, however, there is a heightened necessity to couple the expansion index with snowpack properties and weak layer heterogeneity, as well as to calculate the avalanche release area from the propagation distance.

The energy release rate and specific fracture energy are also crucial parameters in determining the stability and fracture extension of the snowpack. The critical crack length obtained by PST experiments can be employed to calculate the specific fracture energy of the weak layer, which was measured in the range of 0.08 to 2.7 J m⁻² [73]. The fracture energy of the weak layer is observed to increase with increasing density (Figure 10). Nevertheless, current energy release and dissipation models do not consider other forms of energy transformation, such as the internal energy of the snowpack. The potential energy increases the sliding friction between the slab and the weak layer, in addition to providing the fracture and volume deformation of the weak layer. It is also more appropriate for future energy models to account for more ways of converting energy.

For numerical simulation, earlier idealised loading experiments by FEM were used to compute the stress distribution [7] and energy variation under skier loading for the snowpack. But the FEM is based on the continuous medium theory, and this method does not take into account the heterogeneity of the snow layer. The SNOWPACK model [22] can be employed to evaluate the stability of snow slopes and to quantify alterations in variables such as critical crack length over time. This enables the indirect prediction of the formation of avalanches. The DEM [21,26,46] and MPM [36,37] are the primary numerical methods for the investigation of avalanche formation dynamics processes. The motion of the snow layer particles can be accurately tracked using DEM, and the formation and breakage of adhesive bonds between the particles can also reflect the fracture failure and sintering of the snow layer. The MPM is suitable for research dealing with procedures involving large deformations and fractures. By simulating the mechanical properties of different snow layers, DEM and MPM can reproduce the series of initial failure, dynamic crack propagation and slab fracture in the snow layer. The critical crack length, crack propagation speed and propagation distance were similarly derived to change with snowpack parameters and terrain angle. In addition, DEM and MPM can be used to detect variations in the internal contact stress that cannot be observed in experiments (Figure 11). In the supershear regime, the stress undergoes a clear transition from compressive stress-dependent to shear stress-dominated [46,47]. In view of the uncontrollable and dangerous quality of natural avalanches, numerical simulation will be a mainstream way to study avalanche science in the future.

It remains unfeasible to forecast a specific avalanche event with sufficient temporal and spatial precision. Therefore, the primary objective for the future is to provide theoretical support for the establishment of an avalanche early warning system through a better understanding of the mechanics of avalanche science. In light of this, we propose the following outline of future research directions. In the sub-Rayleigh regime: (1) Modelling and numerically considering the effect of a real slab’s stratified structure on crack propagation in the weak layer. (2) Exploring the disparity between upslope and downslope crack propagation, e.g., the variation in the critical crack length. (3) Investigating which snowpack properties contribute to the crack arrest and quantifying the scale of avalanche release.

In the supershear regime: (1) Refining the effect of angle on supershear regimes to capture the critical angle of translation. (2) Quantifying the influence of snowpack parameters, such as elastic modulus and strength, on supershear crack propagation. Subsequently, cross-slope spreading distances and velocities should be studied from real avalanche experiments, which are currently lacking and necessary to determine the size of the release. In addition, by considering the relevance of scale, it is also necessary to analyse the intrinsic connection between the two processes of avalanche formation and avalanche flow and to establish a unified model for the coupling of the two.

Due to the complexity of real environmental conditions, multidisciplinary, multimethod and multifactorial coupling is the foundation for a comprehensive analysis of avalanche science. This work provides new insights into avalanche dynamics and suggests some pathways for future studies of the formation of avalanches. Through a comprehensive understanding of the formation of avalanches and avalanche dynamics, effective avalanche protection measures can be planned in conjunction with the actual terrain. This will provide a useful contribution to avalanche research and avalanche warning.