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

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.