the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Aug 2026
Mach cone refraction in layered snow slabs during supershear fracture
Abstract. When a crack accelerates beyond the material shear-wave speed, shear-waves pile up into a shock front, a striking Doppler-induced signature of supershear fractures. While supershear fracture is best known from dynamic rupture in strike-slip earthquakes, it has recently been observed and modeled in snow slab avalanches triggered by the failure of a weak layer beneath cohesive slabs. These slabs are typically stratified, with layers of distinct mechanical properties that strongly influence wave propagation. Here, we combine numerical simulations with a long snow-fracture experiment to identify Mach cones associated with supershear crack propagation in layered snow slabs. We show that slab layering alters both the crack propagation speed and Mach cone geometry, and that curvature of the cone can be explained by wave refraction at layer interfaces. These findings advance our understanding of avalanche release and supershear fracture dynamics and provide a framework for detecting and interpreting Mach cones in snow-fracture experiments.
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Status: open (until 30 Sep 2026)
- RC1: 'Comment on egusphere-2026-3388', Pascal Hagenmuller, 24 Aug 2026 reply
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General comment :
A crack propagating in a weak layer buried below a cohesive slab is a key process of dry-snow slab avalanche release. It was recently shown (e.g., Trottet et al., 2022; Bergfeld et al., 2025) that, on a slope, crack propagation can transition from a sub-Rayleigh regime (dominated by mode −I failure) to an intersonic regime (dominated by mode II failure). In the latter case, the crack in the weak layer propagates faster than the shear waves it emits, which should produce a Mach-cone signature in the slab deformation field. The authors numerically reproduce this transition and identify the associated Mach cone using an MPM model combining an elastic slab with an elasto-plastic weak layer (Cohesive Modified Cam Clay). The modelling framework closely follows Trottet et al. (2022), apart from the use of Hencky elasticity. The model is applied to schematic stratified slabs and to a test case corresponding to the experiment of Bergfeld et al. (2025). The authors show that stratification influences both the crack propagation speed and the shape of the Mach cone.
The paper is generally clear and easy to follow. The "Nature-like" ordering of sections is pleasant, but the most interesting and clearest figures are, unfortunately, relegated to the appendix. The methodology is sound, and the agreement with the experimental case lends confidence to the numerical results, although the evaluation is limited to a single test case and a very small number of scalar quantities (crack speed, supercritical crack length). The sensitivity study on layering is likewise restricted to three-layer slabs with a constant vertical density gradient. This limited set of configurations appears sufficient to explain the shape of the Mach cone (piecewise linear, controlled by layer-wise shear-wave speeds via refraction), but is frustrating for understanding how layering controls crack speed. In particular, the effective wave speed of Eqs. (4)–(11) is a purely empirical fit, not grounded in any theory, calibrated on this narrow set of configurations and then extrapolated to the experimental profile. This makes the corresponding results appear rather incremental. Broadening the test cases beyond linear density profiles and/or grounding the effective wave speed in existing theory (e.g., effective-medium/laminate theory or the earthquake literature) would substantially strengthen the paper. Provided that this point (detailed below, along with additional technical comments) is addressed, I consider the paper suitable for publication in TC.
Main comment :
The authors observe that the steady-state supershear crack speed in layered slabs deviates from that of the equivalent homogeneous mean slab. They capture this through an empirical correction fitted separately to increasing and decreasing three-layer density profiles (Eq. 5), then generalise it via a linear regression of the density profile (Eqs. 6–11), and apply it to the Bergfeld profile. No theoretical basis is given for the form, the coefficients, or their extrapolation. Moreover, the fit is calibrated on a single profile type (linear gradients), whereas the very studies cited as inspiration for the layered set-up (Habermann et al., 2008; Monti et al., 2016) considered a much wider variety of layering configurations. Such an extension seems to be a legitimate expectation for a paper whose numerical framework and experimental reference are already established elsewhere.
In the homogeneous case, Trottet et al. (2022) numerically showed that the steady-state crack speed locks onto the extensional wave speed of the slab (remark: the numerical set-up is plane strain, while a narrow field PST is closer to plane stress; the extensional speed then involves E/(1-nu**2)). For a layered plate, the extensional wave speed is sqrt(/). It is thus meaningless to compare the effective speed to the average speed (l. 143–145) or to the fastest speed (l.147-149). However, this locking principle cannot be the whole story for layered slabs. Indeed, the extensional wave speed is independent of the layering order, whereas the fitted coefficients depend on the sign of the density gradient. One feels that layers closer to the weak layer (wave source) have a greater impact on crack-speed selection than those farther away. I am not a specialist in the impedance of layered plates, and the authors may find useful pathways in the dedicated literature. Meanwhile, one may simply assume that the influence of the slab layers is weighted by exp(-d/l), where d is the distance to the weak layer, and l is the characteristic length. With l of the order of 2-3 times the slab thickness, I obtain a good agreement with the three-layer test cases and the Bergfeld case. A few simulations with a non-monotonic profile (e.g., all Monti/Habermann schematic cases) would test the formulation where it matters, and would connect to the interesting observation of fluctuating speeds and crack-arrest propensity for γ < 0 (l. 163–165).
Technical comments