Super Collapse 3(Cracked) Game Download
Thechosen Wong
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The release of snow slab avalanches results from a succession of mechanical processes1,2. A failure is initiated in a highly porous weak snow layer buried beneath a cohesive snow slab, leading to mixed-mode and quasi-brittle crack propagation along the slope3. If the slope angle is larger than the weak layer friction angle, the slab eventually slides and releases4.
A simulation is presented of critical (green squares) and super-critical (red circles) crack lengths normalized by the elastic length Λ as a function of the normalized slope angle \((\psi-\phi^*)/\phi\) (ϕ, friction angle; \(\phi^*\), effective friction angle). Solid and dashed lines correspond to theoretical curves for effective friction angles of \(\phi^*=\phi\) and \(\phi^*=0.4\), respectively. The inset shows \(\phi^*/\phi\) as a function of the collapse amplitude h normalized by weak-layer thickness Dwl (data points correspond to an average of several simulations and error bars represent the s.d.).
Numerous numerical simulations were performed for different slope angles and mechanical properties of both the slab and weak layer (Methods) to evaluate the condition for the onset of anticrack (ac) and supershear (asc) propagation. The critical crack length decreases with increasing slope angle ψ and is on the order of 0.1Λ (Fig. 3). The super-critical crack length only exists if the slope angle ψ is larger than the effective friction angle \(\phi ^* \). It also decreases with increasing slope angle and varies between 0.5Λ and 15Λ in the simulations. The effective friction coefficient controls the onset of the supershear transition and significantly depends on the collapse amplitude h of the weak layer (Fig. 3, inset). Without volumetric collapse, the effective friction angle is exactly equal to the friction angle. However, increasing collapse heights reduce the effective frictional resistance of the shear band, as reported in ref. 4. This local friction reduction enables a supershear transition for slope angles lower than the weak-layer friction angle. In effect, once the crack reaches its super-critical crack length, its sharp acceleration is associated with a significant increase of the slab section that is not supported by the weak layer, leading to unstable propagation even below the friction angle. The simulation data are well reproduced by equation (1) for an effective friction angle between 0.4ϕ and ϕ.
Furthermore, our findings indicate that the crack propagation speed measured in small-scale experiments is not necessarily representative of crack speeds on real avalanche terrain in the down/up-slope direction (mode II). In fact, despite the different propagation mechanisms, the experimentally measured values are in good agreement with the avalanche cross-slope propagation speeds (Fig. 4b, mode III), which are theoretically limited by the Rayleigh wave speed23. Here we provide a two-dimensional (2D) theoretical framework that allows the conditions for the onset of this transition to be evaluated, as well as the crack propagation speed, which depends on slab elastic waves. Future work should include slope-scale experiments and simulations to study 3D propagation patterns35,36, as well as the complex interplay between the weak layer and slab fracture during the release process.
Our findings shed light on a previously unreported stage of the avalanche release process, with key implications for predictions of avalanche danger. More generally, our results reinforce the analogy between snow slab avalanches and earthquakes. Although the mechanism of supershear propagation has rarely been reported in large strike-slip earthquakes15, it requires a very common combination of topographical and mechanical ingredients in slab avalanche release.
On 31 January 2019, a professional snowboarder triggered a dry-snow slab avalanche (Extended Data Fig. 4a) in a location near Col du Cou in Wallis, Switzerland (Extended Data Fig. 5a). A few minutes previously, the group had checked the snowpack stability on a slope immediately behind and did not trigger an avalanche (see ski tracks in Extended Data Fig. 4b). The snowboarder triggered this slab avalanche because of the large impact force induced by a jump from the ridge (Supplementary Video 3), which led to failure initiation and crack propagation in the buried weak snow layer close to the ground.
The avalanche was recorded using a high-quality video camera (with a frame rate of 50 frames per second), allowing analysis of slab motion induced by crack propagation within the buried weak snow layer. Our crack propagation speed measurements rely on four stages: (1) video stabilization using optical flow, (2) Eulerian video magnification (EVM) to enhance small changes in snow reflection due to slab deformation, and detection of (3) time and (4) location of slab deformation between video frames. After determining the location and timing of slab deformation, we calculated the deformation distances and time difference from the crack initiation point (impact of the snowboarder) and time and accounted for spatial and temporal uncertainties. This allowed us to drive an estimation of the average crack propagation speed in the determined direction.
Our procedure to evaluate the crack propagation speed based on the EVM method was validated by applying it to (1) 2D and 3D numerical simulations performed on flat and inclined slopes (Extended Data Fig. 6) and (2) classical PST experiments on both flat and inclined slopes and a flat 5-m-long PST31 (Extended Data Fig. 7).
The mechanical properties of snow are time-dependent, and with increasing strain rates the strength decreases while the elastic modulus increases (for example, ref. 41). In addition, the ice matrix in snow has a highly disordered, cohesive-granular microstructure. As such, snow microstructure plays a crucial role in the rich rate- and temperature-dependent behaviour observed in snow (for example, refs. 41,42). It is known that density alone is not sufficient to describe snow mechanical properties, as, for a given density, values scatter by orders of magnitude due to differences in snow microstructure (for example, ref. 43). Furthermore, in slab avalanche release, the snow slab is usually composed of multiple layers with different snow types. To model the complete physics of snow slab avalanche release would thus require accounting for temperature, strain rate, snow density and microstructure. As many of the processes in snow mechanics are still poorly understood, the slab is therefore usually modelled as a homogeneous layer with a bulk density and an effective elastic modulus25. Therefore, to compute the speed of the elastic waves in snow, we used an approximation for the effective elastic modulus of the slab based on density according to the laboratory experiments of Scapozza44, who provides values similar to those measured in PST experiments25. The following power-law relationship was used:
After the supershear transition, the crack propagates in mode II (Fig. 1). We thus describe the onset of supershear transition based on a shear band propagation model to predict the supercritical crack length. The approach used is similar to that in refs. 29,47,48, but introduces the effect of the collapse height on the residual shear friction.
We thank R. Flck for the avalanche photographs. We also thank M. Schaer, L. De Martin and T. Bessire for providing us with additional information and materials regarding the slab avalanche in Col de Cou. We acknowledge S. Mayer for the evaluation of the slab density in the full-scale avalanche based on manual snow profiles and SNOWPACK simulations. We acknowledge J.-F. Molinari for helpful and constructive discussions on the topic of supershear crack propagation. J.G. acknowledges financial support from the Swiss National Science Foundation (SNF; grant no. PCEFP2_181227). G.B. and B.B. were supported by a grant from the Swiss National Science Foundation (200021_169424). C.J. acknowledges financial support from the National Science Foundation (awards nos. 2153851, 2153863 and 2023780) and the Department of Energy (award no. ORNL 4000171342) of the United States.
B.T. performed and analysed the numerical simulations, developed the analytical model, and made all the figures for the paper under the supervision of J.G. and A.v.H. R.S. developed the video analysis method and evaluated the crack propagation speed in the avalanche. G.B. analysed the PST experiments, and A.v.H. designed and performed the PST experiments. C.J. developed the MPM model. J.G. conceived the key idea of the study and developed the original anticrack elastoplastic model, which was later modified by B.T. B.T. wrote the manuscript with J.G., including comments from all co-authors.
a, Schematic of the Propagation Saw Test. b, Histogram of the normalized crack propagation speeds obtained from 222 PSTs following a beta law probability density distribution (solid line). The dashed line corresponds to the median of the distribution.
a, Temporal evolution of the crack tip location. Blue squares and red circles correspond respectively to critical and super critical crack lengths. b, Normalized crack propagation speed as a function of slope angle. The transition to the supershear regime requires a smaller super critical crack length for larger slope angles. A steeper slope induces a shorter transient regime. c, Temporal evolution of the crack tip location. d, Normalized crack propagation speed as a function of the crack length for different values of slope angle. This CPST setup induces propagation in both upslope and downslope directions. While the complex interplay between slab bending induced by weak layer collapse and slab tension (upslope) or slab compression (downslope) breaks the propagation symmetry, the asymptotic speeds are essentially the same in both directions.
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