Propagation of weak layer failure in snow slab avalanche release: analytical solutions for a compliant interface with finite softening
Snow slab avalanches are among the most dangerous hazards in mountain regions. Recent numerical, field, and laboratory studies have renewed interest in shear-failure interpretations of avalanche release, particularly in relation to dynamic crack propagation and supershear fracture. Yet most analytical models either idealize the weak layer as perfectly brittle or neglect its pre-peak elasticity, although post-peak dissipation and compliance both control stress redistribution and critical length. Here, we derive an analytical solution for shear-failure propagation beneath an elastic snow slab with finite linear softening. Building on the weak-spot model of Gaume et al. (2013), failure is described by a fully softened residual core, a fracture process zone, and an intact elastic region. The solution recovers the classical brittle length as softening vanishes, distinguishes the residual crack length from the total affected length, and links weak-spot and fracture-energy descriptions through the softening law. Depth-averaged Material Point Method simulations confirm the analytical stress and displacement profiles and the predicted characteristic lengths. We then extend the same compliant-softening framework to collapse-driven anticrack propagation. A simplified Timoshenko anticrack analogue shows that slab bending and transverse shear deformation amplify normal stress at the weak-layer front and introduce a bending-controlled softening length with an approximately fourth-root dependence on softening displacement, supported by three-dimensional MPM simulations. Finally, a mixed-mode Timoshenko formulation couples weak-layer compression, slope-parallel shear, and slab rotation. A compact sharp-front model and a fully coupled finite-softening model reproduce the observed slope-angle dependence of critical cut length using realistic elastic and failure-envelope parameters.