Growth of shear failure in snow slab avalanche release: analytical solution for a compliant weak layer with finite softening

TL;DR

This paper presents an analytical model for shear failure in snow slab avalanches, incorporating finite softening and elastic mismatch, advancing understanding of fracture propagation in avalanche release.

Contribution

It derives an analytical solution accounting for post-peak softening and elastic mismatch, linking weak-spot and fracture-energy approaches in avalanche modeling.

Findings

Analytical solution distinguishes fully softened crack length and total affected length.

Model recovers classical critical length in the limit of vanishing softening.

Approximate formula relates crack length to softening displacement and critical length.

Abstract

Snow slab avalanches are among the most dangerous natural hazards in mountain areas. Recent progress in numerical modelling, field measurements, and large-scale fracture experiments has renewed interest in shear-failure interpretations of avalanche release, particularly in connection with dynamic crack propagation and supershear fracture. Yet most existing stress-based models either assume a perfectly brittle stress drop, neglecting post-peak energy dissipation, or neglect weak-layer pre-peak elasticity, which influences stress redistribution and critical crack length. Here, we derive an analytical solution for shear-failure propagation in a weak layer beneath an elastic snow slab, explicitly accounting for finite post-peak softening and elastic mismatch between slab and weak layer. Building on the one-dimensional weak-spot framework of Gaume et al.\ (2013), we consider a symmetric failure composed of a fully softened zone, a fracture process zone with linear softening, and an intact elastic region. In the limit of vanishing softening displacement $\delta$, the model recovers the classical stress-based critical length $a_{c0}$. For finite softening, the solution distinguishes between the fully softened crack length $a_c$ and the total affected length $b_c$, which includes the fracture process zone. The formulation provides a direct analytical link between weak-spot and fracture-energy approaches, since fracture energy enters through the constitutive softening law itself. For small softening, the exact solution yields the compact approximation $a_c \simeq a_{c0}\sqrt{1+C_a\delta/u_p}$. This distinction is important when comparing with numerical models that may identify the full damaged region rather than the fully softened zone alone.