ARTICLE https://doi.org/10.1038/s43247-020-00081-8 OPEN Mechanisms of slab avalanche release and impact in the Dyatlov Pass incident in 1959 ✉ Johan Gaume1,2 & Alexander M. Puzrin 3
The Dyatlov Pass incident is an intriguing unsolved mystery from the last century. In February 1959, a group of nine experienced Russian mountaineers perished during a difficult expedition in the northern Urals. A snow avalanche hypothesis was proposed, among other theories, but 1234567890():,; was found to be inconsistent with the evidence of a lower-than-usual slope angle, scarcity of avalanche signs, uncertainties about the trigger mechanism, and abnormal injuries of the victims. The challenge of explaining these observations has led us to a physical mechanism for a slab avalanche caused by progressive wind-blown snow accumulation on the slope above the hikers’ tent. Here we show how a combination of irregular topography, a cut made in the slope to install the tent and the subsequent deposition of snow induced by strong katabatic winds contributed after a suitable time to the slab release, which caused severe non-fatal injuries, in agreement with the autopsy results.
1 School of Architecture, Civil and Environmental Engineering, EPFL, Route Cantonale, 1015 Lausanne, Switzerland. 2 WSL Institute for Snow and Avalanche Research SLF, Flüelastrasse 11, Davos Dorf, Switzerland. 3 Institute for Geotechnical Engineering, ETH Zürich, Stefano-Franscini-Platz 5, 8093 ✉ Zürich, Switzerland. email: [email protected]
COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv 1 ARTICLE COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8
uring the night of February 1, 1959, nine Russian hikers Twenty-six days to three months after the tragedy, search died under unexplained circumstances during a skiing teams found bodies in the forest and on their way back to the D “ expedition in the northern Ural Mountains. The group tent. According to the 1959 Soviet criminal investigation, a had decided to set up their camp on the slope of the Kholat compelling natural force” led to the death of the Dyatlov group. Saykhl (Fig. 1a); the name means “Dead Mountain” in the local However, the nature of this force has not been identified. The Mansi language. Something unexpected happened after midnight mystery arises from numerous unexplained observations. While that caused expedition members to cut the tent suddenly from the hypothermia was determined to be the main cause of death, four inside and escape towards a forest, more than 1 km downslope hikers had severe thorax or skull injuries, two were found with (Fig. 1b), without appropriate clothes, under extremely low missing eyes and one without tongue; some were almost naked temperatures (below −25 °C, see Supplementary Note 3), and in and barefoot, traces of radioactivity were found on some of their the presence of strong katabatic winds induced by the passing of clothes, and signs of glowing orange spheres floating in the sky an arctic cold front1. were reported that night1.
Fig. 1 Pictures taken at the tent location shortly before and weeks after the incident, and a sketch of the slope before the incident. a Last picture of the Dyatlov group taken before sunset, while making a cut in the slope to install the tent. Photograph courtesy of the Dyatlov Memorial Foundation. b Broken tent covered with snow as it was found during the search 26 days after the event. Photograph courtesy of the Dyatlov Memorial Foundation. c Configuration of the Dyatlov tent installed on a flat surface after making a cut in the slope below a small shoulder (see Fig. 2a). Snow deposition above the tent is due to wind transport of snow (with deposition flux Q).
2 COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 ARTICLE
Several theories have been proposed to explain this incident, Here, we show that—even though the occurrence of an ava- including infrasound-induced panic, animals, attacks by Yetis or lanche at this location is unlikely under natural conditions—the local tribesmen, katabatic winds, a snow avalanche, a romantic combination of four critical factors allowed the release of a small dispute, nuclear-weapons tests, etc. The originally popular ava- snow slab directly above the tent. These factors include (Fig. 1c): lanche theory has been questioned due to several contradictory (1) the location of the tent under a shoulder in a locally steeper pieces of evidence1: (1) no obvious signs of an avalanche or debris slope to protect them from the wind, (2) a buried weak snow layer were reported by the search team that arrived 26 days later, (2) parallel to the locally steeper terrain, which resulted in an the average slope angle above the tent location was not suffi- upward-thinning snow slab, (3) the cut in the snow slab made by ciently steep for an avalanche (lower than 30°), (3) the hypo- the group to install the tent, (4) strong katabatic winds that led to thetical avalanche released during the night, at least nine hours progressive snow accumulation due to the local topography after the cut was made in the slope, and (4) the thorax and skull (shoulder above the tent) causing a delayed failure. Furthermore, injuries were not typical for avalanche victims. the possible construction of a parapet1 above the cut (a classical In 2015, the Investigative Committee of the Russian Federation safety procedure to protect the tent from the wind, Fig. SF3) (ICRF) re-opened the investigation and in 2019 concluded that a could have accelerated the failure process. The proposed physical snow avalanche was the most probable cause of the accident2,3. mechanism couples the onset of dynamic shear-fracture propa- The results of this investigation have been challenged recently by gation in the weak snow layer with wind-induced snow transport. the office of the Prosecutor General of the Russian Federation, Provided a realistic wind deposition flux, our model shows that who in 2019 started its own investigation and in July 2020 came the conditions for avalanche release can be met after a delay of 7.5 to the same conclusion as ICRF. Both investigations have not, to 13.5 h from the moment the hikers made the cut in the slope, however, disclosed scientific explanations for the four counter- in agreement with the forensic evaluation of the time of death. arguments listed above and therefore keep being challenged by Dynamic avalanche simulations suggest that even a relatively the relatives, public, and researchers. In particular, a 2019 small slab could have led to severe but non-lethal thorax and skull Swedish-Russian expedition disagreed with the ICRF conclusions, injuries, as reported by the post-mortem examination1. instead proposing that the direct impact of katabatic winds on the tent was the main contributing factor. Results fi 1–9 Based on the signi cant amount of published material ,it Local topography. The mountain slope at the location of the tent seems that previous investigations lack an important ingredient: a is highly irregular. Around 100 m above the tent, there is a quantifiable physical mechanism that can reconcile the avalanche fl fl shoulder which separates a rather at plateau and a steeper slope hypothesis with seemingly con icting evidence. Identifying such a below (Fig. 2a). This slope consists of 4–6 m high steps1 (Fig. 1c mechanism may provide new insights into the nature of storm- and Fig. SF2) and the tent was installed below one of them, where triggered snowpack instabilities, which is another important it was easier to make a cut in a locally flatter slope. The choice of motivation for this work. the tent location was also likely driven by the fact that the larger
a b 60 28° location of the bodies 50
40
30
20 Number of avalanches of Number 10 Shoulder Ravine Tent Cedar 0 10 20 30 40 50 c Slope angle (°) 35 22° Dyatlov Pass 30 25 Holat-Sayakhyl 20
Count 15
10
5
0 5 101520253035404550 Dynamic friction angle (°)
Fig. 2 Local terrain in the Dyatlov Pass, reported snow dynamic friction values, and typical slope angles in human-triggered avalanches. a Terrain map of the Dyatlov Pass—indicated by the red circle (adapted based on JAXA aster data, from USGS Earth Explorer, NASA, USGS, JAXA, CC BY 2.5), blue triangle and black crosses indicate locations of the tent and of the hikers’ bodies, respectively. b Slope angles reported for 139 accidentally triggered avalanches11. c Dynamic friction-angle distribution based on van Herwijnen et Heierli.13. Green = faceted types of crystals (depth hoar, faceted crystals, rounded facets, and surface hoar), black = new snow, decomposed and fragmented crystals, and rounded grains.
COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv 3 ARTICLE COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 scale shoulder would protect them from the strong winds. In and the weak layer10, which can develop under constant loads. It reality, as we show below, this choice of location could have cannot, however, accommodate a nine-hour delay, due to the contributed to the accident: small scale topographic variability relatively short extent of the slab. It follows that the true resulted in a locally steep weak snow layer while the larger mechanism must involve additional loading of the slope. Given shoulder contributed to significant wind-driven snow accumula- the extremely low temperatures and strong katabatic winds, it is tion above the tent, eventually leading to an instability. unlikely that anyone would have climbed above the tent during the night, disturbing the weak layer. In the absence of significant snowfall, the only way to accumulate additional load is through Slope angle and friction. Major arguments against the avalanche wind transport15. hypothesis include insufficient signs of the occurrence of an Snow accumulation above the tent resulted from katabatic avalanche (no apparent deposit or crown fracture) and the rela- winds and the presence of a shoulder located above the tent tively mild slope (~23° ± 2°). It appears, however, that the ground (Fig. 2a). Possible construction of a small snow parapet by the surface in the vicinity of the estimated tent location was steeper Dyatlov group (classical safety measure for snow camping, Fig. (up to 30°) than the average snow slope. Subsequent snowfalls SF3), could contribute to additional loading. smoothed the irregular terrain to the observed average 23° slope. Below, we present an analytical model for a thinning snow slab This implies directly that the slab above the layer of depth hoar gradually loaded by wind-transported snow above the cut in the was thinner uphill, which has three main consequences: (1) It slope, which evaluates the wind deposition flux necessary to results in the average slope of the buried weak layer being about reproduce the forensic estimate of delay. Subsequent numerical 28°, increasing the probability of slab release (Fig. 2b). (2) It modeling confirms that the observed injuries of the Dyatlov reduces tensile support at the top of the slab, considerably lim- group members are consistent with the failed-slab dynamics. iting the avalanche volume. (3) It makes it more difficult to trace avalanche signs, especially 26 days after the incident. While a 28° slope may still be perceived as being too mild to Conditions for a delayed avalanche. The proposed delayed- cause an avalanche—based on the often-quoted and implicit rule release mechanism for a variable-thickness slab (based on the of thumb that a minimum of 30° is required10—in reality, the local topography) is illustrated in Fig. 3. After the expedition critical slope angle can be as low11 as 20° (Fig. 2b), provided the members made the cut in the slope (Fig. 3a), the shear stress in dynamic friction angle (sometimes called crack-face friction11)of the weak layer did not immediately reach the peak shear strength. the snow is sufficiently low. In fact, field measurements have Due to wind-driven snow deposition, the slope gradually became shown that the dynamic friction angle of snow can be as low as loaded over the growing length l , causing the growth of shear 15°, especially for very low temperatures12. In particular, the cw stresses in the weak layer. In addition, partial snow sintering16 buried crystals of depth hoar, which were reported by the leads to strengthening of the wind-transported snow layer. Hence, investigation on the site (see Supplementary Note 3), tend to the growth of l is affected by the gradual increase of the slab exhibit rather low friction values13 (Fig. 2c). cw thickness, and continues until the shear stress in the weak layer τ reaches the peak shear strength p in the vicinity of the cut Additional load due to wind-transported snow. In spite of the (Fig. 3b). A further snow deposition does not mobilize additional fact that the 28° inclination of the buried weak layer was higher shear resistance in the weak layer; in contrast, it initiates rapid than the angle of friction, the snowpack did not fail for at least softening of the weak layer close to the cut, resulting in the loss of nine hours after the slope was cut (see Supplementary Note 1), equilibrium, dynamic uphill growth of the basal shear fracture, due to cohesion in the weak layer. In principle, overcoming and slab release. cohesion does not necessarily require additional loading. Recent The slab release condition is derived in the Methods section as work on delayed snow avalanches14 has proposed a release a solution of the plain strain (2D) boundary value problem of a mechanism driven by rate-dependent processes in the snow slab variable-thickness cohesive slab under a variable-thickness
Fig. 3 The mechanism of delayed release of a wind-induced avalanche. a Geometry: A planar weak layer of thickness d is inclined at an angle α to the h x = h x l horizontal. It is buried below a snow slab of variable depth, decreasing from 0 at the cut ( 0) to c at the straight upper slope ( c). After the cut is made, a wind-transported snow layer of the height hw0 (at the cut) loads the slope, causing local thickening of the slab over the height hs0. b Shear stress in the weak layer.
4 COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 ARTICLE cohesionless snow load, when the shear stress in the weak layer at infinite and zero sintering rates, respectively (Fig. 4b): τ τ the cut 0 reaches the peak strength p: ρ φÀ φ ¼ tan tan min ; hw0; min h0 ρ φ ðÞÀλ φ P w tan max 0 tan ð Þ g0 r3 Pw0 ð Þ ρ φÀ φ 5 τ ¼ τ þ r þ ≥ τ : 1 ¼ tan tan min : 0 g0 1 þ p hw0; max h0 ρ φ ðÞÀλ φ L0 r3 2 L0 w tan max s tan Forces and stresses in Eq. (1) are defined as Finally, we obtain the bounds for the time of delayed release ¼ 1 ρ 2 α; ¼ ρ Á α; (Fig. 4c) by substituting the bounds for hw0 into the equation Pg0 2 K0 gh0 cos Pw0 wghw0L0 sin ! ÀÁ ð2Þ τ ¼ ρgh sin α; τ ¼ ρh þ ρ h g cos α tan φ þ c; ρ h L l 3 g0 0 p 0 w w0 Δt ¼ w w0 0 1 À 1 À c ; ð6Þ 3Q L where φ and c are the angle of internal friction and the cohesion 0 ρ ρ of the weak layer, and w are the densities of the original slab where Q is the snow deposition rate. and of the wind-transported snow, respectively, hw0 is the The proposed analytical framework allowed for back- thickness of the wind-deposited snow at the cut, K0 is the at rest ÀÁpffiffiffiffiffiffiffiffiffiffiffi À calculation of the wind deposition flux Q from the range of fi ¼ À = 1 lateral snow pressure coef cient, L0 lc 1 hc h0 is a possible delays between 9.5 and 13.5 h based on forensic data (see fi characteristic length, with quantities lc, hc and h0 de ned in Supplementary Note 1). For the geometric and physical Fig. 3a. The parameters r1 and r3 are given by parameters of the Dyatlov case (presented in the Methods and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fi þ þ λ þ þ λ discussed in Supplementary Note 6), a reasonable t of the ¼ 1 1 4 0 ; ¼ 1 1 4 s ; ¼ : À1 À1 r1 r3 observed delay range (Fig. 4c) is given by Q 0 008 kg m s . 2 2 ð Þ 15 λ 2 3 Extensive research on snow accumulation shows that this λ ¼ 0h0 ; λ ¼ GL0 ; deposition flux requires average wind velocities in the range of s þ 0 0 − h0 hs0 E h0d 2–12 m s 1 which is in agreement with data from nearby weather fi where the quantities hs0 and d are de ned in Fig. 3a, G is the shear stations on the night of the accident (see Supplementary Note 3). modulus of the weak layer, E′ is the plane strain elastic modulus λ λ of the slab. The numbers 0 and s characterize the ratio between Impact on a human body. In addition to explaining the delay, 14 the geometrical quantity L0 and the characteristic elastic length our proposed mechanism provides the pre-failure slab geometry, of the slab-weak layer system before and after snow transport, which can help in understanding how a relatively small slab respectively. caused the severe but non-fatal injuries reported. We address this Figure 4a, b provides a conceptual framework for calculating question by combining a novel numerical model with existing the time of avalanche release. It has been shown (see Methods) data for human-thorax injuries from impact tests performed by that delayed slab release is only possible when the angle of the automotive industry17,18. internal friction in the weak layer φ falls within the following Three-dimensional numerical simulations based on the range (Fig. 4a): Material Point Method (MPM) and finite-strain elastoplasticity19 tan φ < tan φ < tan φ ðÞλ ; (see Methods) show that this small-slab avalanche impacted the min pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimax 0 hikers lying on the tent floor and filled the excavated space but φ ¼ α À c þ 2α þ K0h0 ; fi tan min tan ρ 1 tan r1 did not have a signi cant runout, consistent with the reported gh0 2L0 ð4Þ pffiffiffiffiffiffiffiffiffi lack of clear avalanche signs (Fig. 5). The predicted length of 5.0 1þ 1þ4λ tan φ ðÞ¼λ pffiffiffiffiffiffiffiffiffi0 tan α; m for tensile failure of the slab is in agreement with the analytical max 0 5þ 1þ4λ 0 model. The simulated snow slab reached a velocity around 2 m/s upon impact (see Fig. SF7, Supplementary Note 5 and For a value of φ within the above range, the shortest and Supplementary Movies 1, 2). At this velocity, an impact on a longest possible delays occur when hw0 reaches the minimum human thorax of a typical snow block with a volume of 0.125–0.5 ð Þ ð Þ fi 3 3 hw0; min and maximum hw0; max critical values de ned by m and density 400 kg/m results in a maximum thorax
Fig. 4 Conceptual framework for calculating the time of avalanche release. a Conditions for delayed slab release. b Bounds for the critical height of the wind-transported load. c Predicted vs. forensically estimated delays.
COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv 5 ARTICLE COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8
Fig. 5 MPM simulation of the dynamics of a snow-slab avalanche and its impact on a human body. Blue snow particles represent regions in which the yield condition has been met. The inset shows a simulated (3D MPM) thorax deflection (normalized by an original thorax height of 20 cm) for a crash test with a 10 kg rigid mass impacting the thorax at a 7 m/s velocity (black curve) and for impacts by 0.125, 0.25, and 0.5 m3 elastoplastic snow blocks of 400 kg/m3 density and velocities of 2 m/s (orange dashed–dotted curve, red dashed curve, and red solid curve, respectively). Injury levels correspond to the AIS11. deformation between 28% and 34%, corresponding to the lower constitutive models with large-deformation dynamic numerical range of values reported from crash tests by Kroell et al.18 for a analysis (MPM) and biomechanical modeling of the human body. 10 kg mass impacting the thorax at a speed of 7 m/s (Fig. 5, This opens new perspectives for research on the effects of snow Supplementary Note 5 and Supplementary Movies 3–6). avalanches on human health and safety. According to the Abbreviated Injury Scale18 (AIS), these Needless to say that our models are based on a number of deflections would mostly lead to non-fatal thoracic injuries from assumptions, which can be justified for this particular case study moderate to severe, in agreement with the autopsy report of the and relaxed for future research. For example, given the very low Dyatlov-incident criminal investigation. Such injuries are not reported temperatures, we assumed a brittle behavior for the usually observed in avalanche victims, because impacts rarely weak layer which allows neglecting the effects of the process occur against stiff obstacles. In the Dyatlov case, the victims were zone30. Moreover, weak layer volumetric collapse19,22,23 did not trapped between the falling slab and the tent floor, which was have to be accounted for in our approach because this layer placed on compacted snow reinforced by skis1. remains completely intact before the onset of instability. Fur- thermore, the analytical model assumes a 2D geometry which in this case can be justified by the fact that the length of the shoulder Discussion controlling snow deposition is much larger than the length of the Significant progress in snow and avalanche research over the past tent. Yet, the 2D profile of the deposited snow has been simplified two decades19–23 has allowed better understanding of avalanche for the sake of obtaining a closed-form solution. An important dynamics and of the processes related to snow-slab avalanche source of uncertainty lies in the dependency of the wind release24–28. Nevertheless, no mechanism similar to the one deposition flux on the average wind velocity. The available inspired by the Dyatlov mystery has been explored in the litera- research15 shows a very wide range of measured deposition fluxes ture, and its physical quantification required new theoretical for a relatively narrow range of the average wind velocity. developments. Nevertheless, the range of wind velocity back-calculated using the In our analytical model, these developments include a snow analytical model and the forensic estimation of the delay is in slab with a spatially variable thickness and its evolution due to good agreement with the range reported in nearby weather sta- sintering of the wind-transported snow, which affects the tions during the night of the accident. instability of a buried weak snow layer. This is highly relevant for Concerning the numerical models, because our main focus was the study of natural storm-triggered slab avalanches29 because its the global thorax response, the skeleton and individual ribs were application is not limited only to wind-blowing snow events but not analyzed. Focusing on the thorax impact, the MPM simula- can also account for additional loads due to a snowfall. The tion is initiated at the onset of slab release and the weak layer is variable slab geometry resulting from irregular local topography not explicitly modeled. In spite of these simplifications, both the and the cut made in the slope play a critical role in determining analytical and numerical models independently predicted a whether or when an instability will occur. Our numerical simu- similar size of the failed slab, providing additional validation for lation of the impact of a snow avalanche on a human body the new mechanism. In addition, while our simulations show that constrained by an obstacle combines advanced elastoplastic in principle, the observed injuries could have resulted from the
6 COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 ARTICLE avalanche impact, the impact-induced deformations of the thorax After the cut is made, the snow keeps accumulating on the slope behind the tent ðÞ; would be rather sensitive to the size of the disintegrated slab due to the wind transport, with hw x t being the thickness of the wind- blocks (Fig. 5) and thus to the relative positions of the bodies with transported snow at the time t after the construction of the parapet. Assuming that the surface of the new snow layer can also be described by a parabolic function, the respect to the cut and slope direction. Given this uncertainty, it is thickness of the wind-transported snow is 1 also possible that the thorax injuries were the result of a later 2 ðÞ¼; ðÞþ; ðÞ¼; ðÞðÞþ ðÞ À x ; snow impact in a very steep ravine where the bodies of the vic- hw x t hs x t hf x t hs0 t hf0 t 1 for x COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv 7 ARTICLE COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 At the moment of the cut, the thickness of the slab is given by Eq. (7), therefore forces: τ ¼ 0, and differentiating Eq. (12) with respect to x and substituting Eq. (16) into pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2P ÂÃ1 þ 1 þ 4λ it gives a differential equation for net lateral forces: ΔPðÞ¼x w0 ðÞ1 À x r3 ÀðÞ1 À x 3 ; r ¼ s : ð30Þ λ À 6 3 2 ∂2Δ s ðÞÀ 2 P ¼ λ Δ ≤ ≤ ; ð Þ This leads to the following normal forces in the slab and shear stresses in the 1 x 0 P for 0 x 1 18 ∂x2 weak layer: where ðÞ¼ ðÞþ Δ ðÞ P x Pc x ÂÃP x rffiffiffiffiffiffiffiffiffiffiffi r 2P r 3 2 0 ¼ ðÞÀ ðÞÀ 1 þ w0 ðÞÀ 3 ÀðÞÀ ; x L E h d Pg x Pg0 1 x λ À6 1 x 1 x x ¼ ; λ ¼ 0 ; l ¼ 0 : ð19Þ s 0 2 e0 τðÞ¼ τ ðÞþ τ ðÞþ 1 ∂ΔP L0 le0 G x c x w x ∂ L0 x ð31Þ – À Equation (18) is the well-known second-order Euler Cauchy differential ¼ τ ðÞþ r1 ðÞÀ r1 1þ Pw0 ðÞÀ 2 g x Pg0 1 x 1 x equation, with a solution given by hiL0 L0 À À 2Pw0 r3 ðÞÀ r3 1À 3 ðÞÀ 2 : r1 r2 λ À 1 x 1 x ΔPðÞ¼x C ðÞ1 À x þC ðÞ1 À x ; s 6 L0 L0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ ¼ λ ¼ 2 À 1 þ 1 þ 4λ 1 À 1 þ 4λ 20 At the cut x 0, using s r3 r3 we obtain r ¼ 0 ; r ¼ 0 ; 1 2 2 2 r P λ À 2r r P r P ¼ 0; τ ¼ τ þ 1 P þ w0 s 3 ¼ τ þ 1 P þ w0 3 ð32Þ 0 0 g0 L g0 L λ À 6 g0 L g0 L r þ 2 where C1 and C2 are constants to be found from boundary conditions. In order to 0 0 s 0 0 3 fi ¼ have a nite solution at x 1, C2 has to be set to zero, and from the boundary and when h ¼ h þ h becomes sufficiently large for this shear stress to reach w0 f0 s0 ¼ Δ ¼À the peak strength: condition at x 0, where the lateral force is equal to zero ( P Pg ) due to x¼0 P the cut: τ ¼ τ þ g0 þ r3 Pw0 ≥ τ ¼ τ þ τ ð Þ 0 g0 r1 þ p p0 pw 33 L0 r3 2 L0 1 2 ΔPj¼ ¼ C ¼ÀP ; P ¼ K ρgh cos α: ð21Þ x 0 1 g0 g0 2 0 0 the slab will fail. Shear stresses and normal forces in Eq. (33) are defined as Substituting these constants into Eq. (20) provides stress conditions in the slab P ¼ 1 K ρgh2cos α; P ¼ ρ ghðÞþ h L Á sin α; τ ¼ ρgh sin α; g0 2 0 0 w0 w s0 f0 0 g0 0 ð Þ and the weak layer after the cut in the slope: τ ¼ ρ α φ þ ; τ ¼ ρ ðÞþ α φ: 34 p0 gh0cos tan c pw wghs0 hf0 cos tan ðÞ¼ ðÞþ Δ ðÞ¼ ðÞÀ ðÞÀ r1 ; Pc x Pg x P x Pg x Pg0 1 x Relationship (33) has been validated numerically for the Dyatlov case ð Þ 1 ∂ΔP r À 22 parameters using the Material Point Method (see Supplementary Note 4). τ ðÞ¼ τ ðÞþ ¼ τ ðÞþ 1 ðÞÀ r1 1 c x g x ∂ g x Pg0 1 x L0 x L0 Time to failure with the boundary conditions at the cut x ¼ 0: . The time to failure is a function of the area Aw of the wind- transported snow at failure derived by integration of Eq. (9) and of the snow r j ¼ ; τ j ¼ τ þ 1 : ð Þ deposition rate Q, which is a function of the average wind speed v: Pc x¼0 0 c x¼0 g0 Pg0 23 ! L0 ρ A 1 l 3 For the slab not to fail immediately after the cut the following condition should Δ ¼ w w ; ¼ À À c ; ð Þ t ðÞhi Aw hw0L0 1 1 35 be satisfied: Qv 3 L0 τ j ¼ τ þ r1 τ ; where the small area of new snow between lc and lcw in Fig. 3a has been neglected. c x¼0 g0 Pg0< p0 L0 ð24Þ After the wind-transported snow sinters, it causes the thickening of the original τ ¼ σ φ þ ¼ ρ α φ þ ; p0 ntan c gh0cos tan c slab, which constrains the slab displacements and slows down the growth of shear stresses in the weak layer. Therefore, if the snow sinters immediately, the peak where φ and c are the angle of internal friction and cohesion in the weak layer. strength in the weak layer is reached slower. It follows that the largest amount of In this case, the normal force and shear stress from Eq. (22) will serve as initial the wind-transported snow h ; required for the avalanche release can be cal- conditions for the next stage—loading of the slab by the wind- w0 max transported snow. culated assuming that the snow sinters instantaneously over the entire thickness hw, ¼ i.e. hf0 0. The upper bound of delay can then be expressed as ! 3 Failure of the undercut slab loaded by the wind-transported snow ρ h ; L . After the Δ ¼ w w0 max 0 À À lc ; ð Þ tmax 1 1 36 wind-transported snow reached the thickness hw given by Eq. (9), the additional 3QvðÞhi L τ 0 shear stress w in Eq. (12) is now larger than zero: where h ; ¼ h is determined by solving numerically Eq. (33), with stresses P w0 max s0 τ ðÞ¼x ρ gh ðÞÁx sin α ¼ w0 ðÞ1 À x 2; and forces substituted from Eq. (34): w w w ð Þ L0 25 ρ φ À φ tan tan min ¼ ρ ðÞþ Á α; h ; ¼ h ¼ h ; ð Þ Pw0 wghs0 hf0 L0 sin w0 max s0 0 ρ φ ðÞλ À φ 37 w tan max s tan ρ where w is the density of the wind-transported snow. The portion hs of the new where snow has sintered, increasing the thickness of the original slab, as reflected in Eq. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ ðÞ c 2 K0h0 (17). After substituting x into Eq. (12), differentiating it with respect to x and tan φ ¼ tan α À ρ 1 þ tan α þ r ; w min gh0 1 2L0 combining it with Eq. (17) we obtain the following differential equation for net pffiffiffiffiffiffiffiffiffi ð38Þ þ þ λ λ r3 1 1 4 s 0 h0 lateral forces in the sintered slab: tanφ ðÞ¼λ þ tan α; r ¼ ; λ ¼ þ : max s r3 2 3 2 s h0 hs0 ∂2ΔP Using the same reasoning, it follows that the smallest amount of the wind- ðÞÀ 2 ¼ λ Δ þ ðÞÀ 3 ≤ ≤ ; ð26Þ 1 x s P 2Pw0 1 x for 0 x 1 transported snow required for the avalanche release h ; can be calculated ∂x2 w0 min ¼ – assuming that the new snow does not sinter. Substituting hs0 0 into Eqs. (37) where (38) gives the equation for the lower bound of the thickness of the wind-deposited x λ h snow at which the slab will fail: x ¼ ; λ ¼ 0 0 : ð Þ s þ 27 ρ tan φ À tan φ L0 h0 hs0 ¼ ¼ min ; ð Þ hw0; min hf0 h0 ρ φ ðÞÀλ φ 39 Solution of Eq. (26) is given by w tan max 0 tan 2P where Δ ðÞ¼ ðÞÀ r3 þ ðÞÀ r4 À w0 ðÞÀ 3; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P x C1 1 x C2 1 x 1 x λ À 6 φ ¼ α À c þ 2α þ K0h0 ; s tan min tan ρ 1 tan r1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð28Þ gh0 2L0 þ þ λ À þ λ pffiffiffiffiffiffiffiffiffi ð40Þ 1 1 4 s 1 1 4 s 1þ 1þ4λ r ¼ ; r ¼ ; φ ðÞ¼λ r1 α; ¼ 0 ; 3 4 tan max 0 þ tan r1 2 2 r1 2 2 fi ¼ where C1 and C2 are constants. Again, in order to have a nite solution at x 1, C2 which, when substituted into Eq. (35), gives the lower bound for the time to failure: ¼ ! has to be set to zero, and from the boundary condition at x 0, where the lateral 3 ρ h ; L force is equal to zero due to the cut: Δ ¼ w w0 min 0 À À lc : ð Þ tmin ðÞhi 1 1 41 3Qv L0 2Pw0 ΔPj¼ ¼ C À ¼ 0 ð29Þ fi fi x 0 1 λ À 6 For a delayed release to be possible, two conditions have to be satis ed. The rst s one is that the slab does not fail immediately after the cut, which according to Eq. substituting these constants into Eq. (28) provides a solution for the net normal (24), after substitution of forces and stresses from Eq. (34) imposes the following 8 COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 ARTICLE restriction on friction: simulate snow and avalanche mechanics. We recall here the main characteristics of φ φ; ð Þ the model. tan min < tan 42 The mass and momentum balance equations are solved using the Material Point φ fi Method (MPM)35 and finite-strain elastoplasticity. MPM is a hybrid Eulerian- where tan min is de ned in Eq. (38). For Eqs. (37) and (39) this implies that that delayed release is only possible when Lagrangian method which is well suited to studying problems involving large deformations, collisions and fractures. It is thus suitable to analyzing the impact of φ φ φ ðÞλ ≤ φ ðÞλ ; ð Þ “ ” tan min < tan < tan max s tan max 0 43 a snow slab on a human body. Particles are used to track position, velocity and fi φ fi deformation gradients and a xed background mesh is used to solve the balance where tan max is de ned in Eqs. (38) and (40). equations. Here we used the Affine Particle In Cell (APIC) method for the transfer between grid and particles36 which allows an exact conservation of momentum and The Dyatlov case. The following parameters for the Dyatlov case have been angular momentum. In addition, we use cubic B-splines as shape functions which adopted (see Supplementary Note 6): guarantees the continuity of the gradients at the nodes. More details about the fi 19 α ¼ ; ¼ : ; ¼ : ; ¼ : ; ¼ : ; MPM model and nite-strain elastoplasticity can be found in Gaume et al. . 28 h0 0 5m hc 0 1m hp 0 5m lp 0 5m A mixed-mode yield surface20 defined in the space of the p-q invariants of the stress ¼ ðÞ: ; ¼ : ; ¼ : À2; fi ¼À ðτÞ= τ lc 4m 216 slope at the cut le0 1 0m g 9 81 ms tensor is used. The pressure p is de ned as p tr 3where is the Kirchhoff À À fi ¼ð = : Þ1=2 ρ ¼ 400 kg m 3; ρ ¼ 300 kg m 3; K ¼ 0:5: stress tensor. The von Mises equivalent stress q is de ned as q 3 2 s s with w 0 ¼ τ þ 20 s pI the deviatoric stress tensor (I: identity matrix). The cohesive Cam Clay A reasonable value of the friction in the weak layer (see Supplementary yield surface is given by Note 6) is φ ¼ 20. This sets the upper bound of cohesion, for which a finite h ; can still be determined, at c 440 Pa. This is well within the range ðÞ¼; ðÞþ β 2 þ 2ðÞþ β ðÞ¼À ; ð Þ w0 max yp q 1 2 q M p p0 p p0 0 47 reported by Jamieson and Johnson33, and from Eqs. (37)–(40) we obtain in which p is the pre-consolidation pressure, M is the slope of the critical state line and ¼ : ; ¼ : : 0 hw0; min 0 24 m hw0; max 0 44 m β controls the resistance to tension. At the end of the stress increment, if ypðÞ; q < 0the À À ’ ’ ’ By adjusting the deposition mass flux to Q ¼ 0:008 kg m 1 s 1, from Eqs. (36) material is elastic and follows Hooke slaw(withaYoungs modulus E and Poisson s ν – and (41) we determine ratio ) implemented in the framework of hyperelasticity with a St Venant Kirchhoff model with Hencky strain37. Δ ¼ : ; Δ ¼ : ; tmin 7 2h tmax 13 5h If the snow fails, hardening or softening is simulated by expanding or shrinking the yield surface, respectively, through variations in p . Compression leads to which overlaps with the forensically estimated range of Δt ¼ 9:5 À 13:5 h (see 0 hardening, promoting compaction, while tension leads to softening, promoting Supplementary Note 1). Extensive research on snow accumulation shows15 that fl – −1 fracture. The hardening law is formulated as a function of the volumetric plastic this deposition ux requires average wind velocities in the range of 2 12 m s ϵP strain V according to which is in agreement with data from nearby weather stations on the night of the ÀÁÀÁ accident (see Supplementary Note 3). The possible range of friction angles in Eq. p ¼ K sin h ξmax ÀϵP ; 0 ; ð48Þ (43) leading to delayed slab release is shown in Fig. 4a as a function of the slope 0 V inclination. Figure 4b shows the corresponding range of the critical heights of the in which K is the bulk modulus and ξ is the hardening factor. wind-transported snow load that cause a delayed release for two cases: with and We model the human body as an hyperelastic solid (St Venant–Kirchhoff without sintering, described by Eqs. (37) and (39), respectively. Using Eq. (35), the model with Hencky strain37). The human body is simulated as a bulk solid with times of delay are plotted as a function of the height of the wind-transported snow material properties taken from thorax impact tests performed by the automotive in Fig. 4c. industry18. This approach allowed us not to simulate all individual bones and organs of the human body. Dimensions of the failed slab. After the shear fracture propagated into the weak The geometry of the slope is taken to be the same as in the analytical model (Fig. 3). The bed surface satisfies a slip boundary condition. The thinning layer, the slab may fail in tension at the distance lt from the cut, over the width B parallel to the cut. Here we determine the width B of the failed slab, assuming that snowpack and the size of the wind-transported slab were also implemented the tension crack formed at the boundary of the wind-transported load defined in according to the analytical model (see Section 1 of the Methods). The human body Eq. (9): of a male of average size (1.70 m) lying on its back was simulated on a rigid bed sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! surface. fi h In order to calibrate our model, we rst simulate the impact of a 10 kg rigid l ¼ l ¼ L 1 À c ; block (0.15 × 0.15 × 0.06 m) moving with velocity 7 m/s on a 3D human thorax t cw 0 h þ h þ h 0 s0 f0 with restrained back (see Supplementary Note 5 and Supplementary Movie 3). This where the thickness of the slab is h . The areas A and A of the initial and sintered setup corresponds to the automotive crash experiments performed by Kroell c 0 s 18 slab cross-sections between the cut and the crack are derived by integration of Eqs. et al. . We adjust the elastic modulus of the body in order to reproduce the same fl (7) and (9), respectively: maximum normalized de ection of 0.49 that resulted in fatal injuries. Next, we perform a high-resolution 2D avalanche simulation using the main features known 3 fi ¼ 1 À À lc þ ðÞÀ ; about the Dyatlov tent con guration (Fig. 1) so as to evaluate the snow slab impact A0 h0L0 1 1 hc lcw lc 3 L0 velocity (up to 2 m/s, see Supplementary Note 5 and Supplementary Movie 2) and ð44Þ 3 3 the range of typical snow block dimensions (up to 0.5 m , see Supplementary l A ¼ 1 ðÞh þ h þ h L 1 À 1 À cw Note 5 and Supplementary Movies 4–6). Next, we simulated the impact of snow s 3 0 s0 f0 0 L0 blocks of 0.125, 0.25, and 0.5 m3, 400 kg/m3 density and impact velocity 2 m/s and the equilibrium at the onset of the tension crack formation is given by (same mechanical properties as the wind-slab) on a human thorax with restrained ÀÁ back. Finally, the severity of injuries was determined by relating the maximum ρ þ ðÞÀ ρ αðÞ¼α À φ σ þ σ : ð Þ A0 As A0 w Bg cos tan tan hcB t 2lcwhc s 45 normalized deflections to the Abbreviated Injury Scale (AIS, see Supplementary In Eq. (45) the left-hand side represents the net driving force acting on the slab, Note 6) presented by Kroell et al.18. while the right-hand side is the sum of the tensile resistance at the back of the slab The following parameters for the Dyatlov case have been adopted (see σ σ and the shear resistance at the sides of the slab, with t and s the tensile and shear Supplementary Note 6): strength, respectively. It is assumed that the contribution of the wind-transported α ¼ 28 ; h ¼ 0:5m; h ¼ 0:1m; h ¼ 0:5m; l ¼ 5:0m: snow to side resistance can be neglected, and the slab is thinning at the sides to hc. 0 c w wc Equation (45) can be rewritten to determine the width of the failed slab B: The simulations were performed with a background mesh size dx ¼ 0:005 m; 4 2l h σ particles per element in 2D and 8 particles per element in 3D. B ¼ ÀÁcw c s : ð46Þ We used the following snow and bed surface mechanical properties for the A ρ þ ðÞA À A ρ g cos αðÞÀtan α À tan φ h σ 0 s 0 w c t Dyatlov case (see Supplementary Note 6): For the parameters of the Dyatlov case: ρ ¼ 300 kg mÀ3; E ¼ 8 MPa; ν ¼ 0:3; φ ¼ ; σ ¼ : ; σ ¼ : ; ¼ : ; ¼ : ; ¼ : ; ¼ : 20 t 6 0 kPa s 5 2 kPa lc 4 0m hc 0 1m h0 0 5m hw0 0 5m β ¼ : ; ¼ : ; ¼ ; ξ ¼ ; 0 2 M 0 8 p0 30 kPa 5 ¼ ¼ : ¼ : À the tension crack can form at lt lcw 4 95 m resulting in B 8 8 m, consistent ρ ¼ 400 kg m 3; E ¼ 17 MPa; ν ¼ 0:3; β ¼ 0:1; = 34 = – w w w w with the observed B lt ratios and comparable to the width of the cut b 6.5 7.5 m ¼ : ; ¼ ; ξ ¼ ; (for a 4.5 m long tent, the cut had to be around 1.0–1.5 m longer on each side). Mw 0 8 p0w 100 kPa w 5 φ ¼ : bed 22 Modeling the impact of the avalanche using the Material Point Method.We As mentioned above, mechanical properties of the human body are chosen simulate the snow slab dynamics and snow block impacts using the model recently based on thorax impact tests performed on human cadavers with restrained back18 developed by Gaume et al.19. This model uses the Material Point Method, finite- to match the corresponding maximum deflection of 0.49 obtained upon impact of a strain elastoplasticity and a Cohesive Cam Clay constitutive relationship to 10 kg rigid mass with 7 m/s impact velocity. We used the following elastic modulus COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv 9 ARTICLE COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 and Poisson’s ratio (see Supplementary Note 6): 23. Barraclough, T. W. et al. Propagating compaction bands in confined compression of snow. Nat. Phys. 13, 272–275 (2017). E ¼ 0:215 MPa; ν ¼ 0:35: hb hb 24. Schweizer, J., Reuter, B., van Herwijnen, A., & Gaume, J. Avalanche release 101. In (ed. Greene, E.). Proc. International Snow Science Workshop (1–11) Data availability (Montana State University, Bozeman, USA 2016). All data needed to evaluate the conclusions in the paper are present in the manuscript 25. Gaume, J., van Herwijnen, A., Chambon, G., Wever, N., & Schweizer, J. Snow fracture in relation to slab avalanche release: critical state for the onset of crack and the electronic supplementary material. Data supporting the plots of the manuscript is – available at https://doi.org/10.5281/zenodo.4302235 propagation. Cryosphere 11, 217 228 (2017). 26. van Herwijnen, A. et al. Estimating the effective elastic modulus and specific fracture energy of snowpack layers from field experiments. J. Glaciol. 62, Code availability 997–1007 (2016). The code necessary to compute the delay for slab avalanche release and avalanche size 27. Sovilla, B., Schaer, M., Kern, M., & Bartelt, P. Impact pressures and flow can be found at https://doi.org/10.5281/zenodo.4088052. A description of the MPM regimes in dense snow avalanches observed at the Vallée de la Sionne test site. model can be found in a previous publication at https://www.nature.com/articles/s41467- J. Geophys. Res.: Earth Surf. 113. https://doi.org/10.1029/2006JF000688 (2008). 018-05181-w. 28. Gaume, J., Chambon, G., Eckert, N. & Naaim, M. Influence of weak-layer heterogeneity on snow slab avalanche release: application to the evaluation of avalanche release depths. J. Glaciol. 59, 423–437 (2013). Received: 14 July 2020; Accepted: 8 December 2020; 29. Bair, E. H., Simenhois, R., Birkeland, K. & Dozier, J. A field study on failure of storm snow slab avalanches. Cold Regions Sci. Technol. 79,20–28 (2012). 30. McClung, D. M. Shear fracture precipitated by strain softening as a mechanism of dry slab avalanche release. J. Geophys. Res.: Solid Earth 84, 3519–3526 (1979). 31. Stott, T. in Outdoor Recreation (187–214) (Palgrave Macmillan, Cham) References https://doi.org/10.1007/978-3-319-97758-4_8 (2019). 1. Buyanov, E. V. and Slobtsov, B. E. Mystery of the Dyatlov Group Death (Taina 32. Jaky, J. The coefficient of earth pressure at rest. J. Soc. Hungarian Architects Gibeli Gruppy Dyatlova). Litres (2014) ISBN 978-5040256921. Eng. 355–358. https://doi.org/10.1139/t93-056 (1944). 2. Ilyshina, M. and Kolirin, L. Russia reopens investigation into 60-year-old 33. Jamieson, B. & Johnston, C. D. Evaluation of the shear frame test for weak Dyatlov Pass mystery. CNN. https://edition.cnn.com/2019/02/04/europe/ snowpack layers. Ann. Glaciol. 32,59–69 (2001). dyatlov-pass-incident-scli-intl/index.html (2019). 34. McClung, D. M. Dimensions of dry snow slab avalanches from field 3. Dyatlovpass.com. Section Timeline (1959). measurements. J. Geophys. Res.: Earth Surf. 114. https://doi.org/10.1029/ 4. Eichar, D. Dead Mountain: The Untold True Story of the Dyatlov Pass Incident 2007JF000941 (2009). (Chronicle Books, San Francisco, 2013). 35. Sulsky, D., Chen, Z., & Schreyer, H. L. A particle method for history- 5. McCloskey, K. Mountain of the Dead: The Dyatlov Pass Incident (The History dependent materials. Comput. Methods Appl. Mech. Eng. 118, 179–196 (1994). Press Ltd, 2013). 36. Jiang, C., Schroeder, C. & Teran, J. An angular momentum conserving affine- 6. Osadchuk, S., & O’Flynn, K. The Dyatlov Pass Incident 30–35 (Fortean Times, particle-in-cell method. J. Comput. Phys. 338, 137–164 (2017). 2009). 37. Jiang, C., Schroeder, C., Teran, J., Stomakhin, A., & Selle, A. The material 7. Kurakin, D. The cultural mechanics of mystery: structures of emotional point method for simulating continuum materials. In ACM SIGGRAPH 2016 attraction in competing interpretations of the Dyatlov pass tragedy. Am. J. Courses (1–52) (2016). Cultural Sociol. 7, 101–127 (2019). 8. Ardashev, R. G., & Turkova, V. N. Ob oshibochnych I nadumannych versiyach pri rassledovaniyach massovoi gibeli lyudei (Po materialam o gibeli Acknowledgements turisticheskoi gruppy Dyatlova). Zakon i pravo. The authors acknowledge Prof. Itai Einav of the University of Sydney for constructive 9. Haynes, E. Can Science Solve the Mysteries of the Dyatlov Pass? (T Brand discussions about our modeling approach, Prof. Chenfanfu Jiang from the University of Studio, The New York Times). https://www.nytimes.com/paidpost/ Pennsylvania for his contribution to the development of the MPM tool and Dr. Alec van facebookwatch/limetown-dyatlov.html (2019). Herwijnen from the WSL Institute for Snow and Avalanche Research SLF (Davos, 10. Schweizer, J., Bruce Jamieson, J., & Schneebeli, M. Snow avalanche formation. Switzerland) for providing us with the snow friction data. The authors also appreciate the Rev. Geophys. 41. https://doi.org/10.1029/2002RG000123 (2003) help of Paul Berclaz and Lars Blatny in preparation of some of the figures of the 11. Vontobel, I., Harvey, S. & Purves, R. S. In (eds Naaim-Bouvet, F., Durand, Y. supplement and for additional model developments as well as insightful discussions with & Lambert, R.) Terrain analysis of skier-triggered avalanche starting zones. In Dr. Francesco Comola on wind-blowing snow. J.G. and A.M.P. acknowledge financial Proc. International Snow Science Workshop ISSW, Grenoble, France, 7–11 support from the Swiss National Science Foundation (grant numbers PCEFP2_181227 October 2013 (Montana State University, Bozeman, USA 2013). and 200021_168998, respectively). We acknowledge Dr. Pascal Hagenmüller and Dr. 12. Casassa, G., Narita, H. & Maeno, N. Shear cell experiments of snow and ice Dieter Issler for constructive comments that contributed to improve our paper. We friction. J. Appl. Phys. 69, 3745–3756 (1991). gratefully acknowledge valuable information and permission to use original drawings 13. van Herwijnen, A., & Heierli, J. Measurement of crack‐face friction in from Evgeniy Buyanov of St. Petersburg, Russia, as well as the Dyatlov Memorial Fun- collapsed weak snow layers. Geophys. Res. Lett. 36. https://doi.org/10.1029/ dation for permission to use photographs. 2009GL040389 14. Puzrin, A. M., Faug, T. & Einav, I. The mechanism of delayed release in Author contributions earthquake-induced avalanches. Proc. R. Soc. A 475, 20190092 (2019). J.G. conceived the key idea of the study, performed the numerical analysis and co-wrote 15. Sturm, M. & Stuefer, S. Wind-blown flux rates derived from drifts at arctic the manuscript. A.M.P. developed the analytical model of delayed avalanche release and snow fences. J. Glaciol. 59,21–34 (2013). co-wrote the manuscript. 16. Podolskiy, E. A., et al. Healing of snow surface-to-surface contacts by isothermal sintering. Cryosphere 8, https://doi.org/10.5194/tc-8-1651-2014 (2014). Competing interests 17. Nahum, A. M., Gadd, C. W., Schneider, D. C., & Kroell, C. K. Deflection of the The authors declare no competing interests. human thorax under sternal impact. In Paper No. 700400 Presented at SAE International Automobile Safety Conference, Detroit. https://doi.org/10.4271/ 700400 (1970) Additional information 18. Kroell, C. K., Schneider, D. C., & Nahum, A. M. Impact tolerance and Supplementary information is available for this paper at https://doi.org/10.1038/s43247- response of the human thorax II (No. 741187). SAE Technical Paper. https:// 020-00081-8. doi.org/10.4271/741187 (1974) 19. Gaume, J., Gast, T., Teran, J., van Herwijnen, A. & Jiang, C. Dynamic Correspondence and requests for materials should be addressed to A.M.P. anticrack propagation in snow. Nat. Commun. 9,1–10 (2018). 20. Reiweger, I., Gaume, J. & Schweizer, J. A new mixed‐mode failure criterion for Peer review information Primary handling editors: Heike Langenberg weak snowpack layers. Geophys. Res. Lett. 42, 1427–1432 (2015). 21. Schweizer, J. On recent advances in avalanche research. Cold Region Sci. Reprints and permission information is available at http://www.nature.com/reprints Technol. 144,1–5 (2017). ’ 22. Heierli, J., Gumbsch, P. & Zaiser, M. Anticrack nucleation as triggering Publisher s note Springer Nature remains neutral with regard to jurisdictional claims in fi mechanism for snow slab avalanches. Science 321, 240–243 (2008). published maps and institutional af liations. 10 COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv COMMUNICATIONS EARTH & ENVIRONMENT | https://doi.org/10.1038/s43247-020-00081-8 ARTICLE Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. © The Author(s) 2021 COMMUNICATIONS EARTH & ENVIRONMENT | (2021) 2:10 | https://doi.org/10.1038/s43247-020-00081-8 | www.nature.com/commsenv 11