Scaling laws for rockfall impact fragmentation emerging from diverse lithologies

Abstract

Impact-induced fragmentation is a fundamental dissipative process in geosciences, yet its stochastic nature makes predicting debris evolution a persistent challenge. Here, we introduce a discrete element framework to resolve fragmentation mechanics across a diverse lithological spectrum, from high-strength siliciclastic units to massive carbonates, validated against high-resolution field data from documented rockfall events. Our results reveal that, despite the inherent randomness of impact dynamics, fragment size distributions consistently follow a universal Weibull scaling law, independent of lithology or initial kinetic energy. By applying a relative breakage index, we demonstrate a remarkable collapse of fragmentation data onto a single statistical signature, bridging the gap between grain-scale fracture and macroscopic debris evolution. We find that this Weibullian signature acts as a proxy for lithological sensitivity, reflecting distinct efficiencies in converting kinetic energy into new fracture surfaces. This framework explicitly resolves the energy partitioning between surviving blocks and comminuted debris, providing a robust predictive link between impact mechanics and structural resilience. From an engineering perspective, our findings enable a shift from idealised single-block impact assumptions toward a realistic assessment of distributed energy in fragmented particle clouds, offering a physical basis for optimising protective galleries and hazard mitigation strategies in complex mountainous terrains.

Figures (11)

• Figure 1: General diagram of a single rockfall on steep slopes. In this schematic, $h$ denotes the impact height of the in-situ rock block, and $D$ its initial characteristic size. The rockfall sequence begins with the recognition of the unstable block on the slope 1. Once it detaches from the face of the rock mass it falls freely and impacts the sediment layer 2. Depending on the degree of energy of the collision, this impact can generate the rebound of the impacting block, or its fragmentation into several fragments 3, where each one can follow a dispersion of different trajectories downstream, depending on the mechanical energy of the system, and experience new fragmentation events.
• Figure 2: Reported rockfall events on which this study is based Ruizcarulla2016Gili2022. (a) Location of events in Catalonia, Spain. (b) Rockfall in resistant cemented sandstone, Lluçà. (c) Limestone open pit in Vallirana. (d) Event in sedimentary units in Els Omells de Na Gaia.
• Figure 3: Calculation scheme of the fast-breakage numerical model. a Work sequence of the particle fracture model. b Visual diagram of the replacement fragmentation process of a smoothed spherical particle.
• Figure 4: Quantification of breakage based on the areas enclosed by the different size distribution curves Einav2007.
• Figure 5: a Irregular rock geometry used in the study. b DWT scheme for numerical calibration tests. c Diagram of the rockfall used in the numerical model, based on the free fall of a rock block impacting onto a granular bed.