RESEARCH ARTICLE A depth average SPH model including μ(I) rheology and crushing for rock avalanches Alberto Longo 1 | Manuel Pastor 2 | Lorenzo Sanavia 1 | Diego Manzanal 2,3 | Miguel Martin Stickle 2 | Chuan Lin 4 | Angel Yague 2 | Saeid Moussavi Tayyebi 2 1 DICEA, Università degli Studi di Padova, Padova, Italy 2 Department of Applied Mathematics, ETSI Caminos, Universidad Politécnica de Madrid, Madrid, Spain 3 Engineering Faculty, INTECIN‐ CONICET‐UBA‐UNPSJB, Buenos Aires, Argentina 4 College of Water Conservancy and Hydropower, Hohai University, Nanjing, China Correspondence Diego Manzanal, Department of Applied Mathematics, ETSI Caminos, Universidad Politécnica de Madrid, Madrid , Spain. Email: d.manzanal@upm.es Funding information Ministerio de Economía, Industria y Competitividad (MINECO); Advanced Modeling of Landslides, Grant/Award Number: BIA2016‐76253‐P; National Key Research and Development Program of China, Grant/Award Number: 2016YFC0401601; Special Fund for Public Welfare Industry of Ministry of Water Resources of China, Grant/Award Number: 201501033; Postgraduate Research & Prac- tice Innovation Program of Jiangsu Prov- ince, Grant/Award Number: KYZZ16_0282 Summary Classical depth‐integrated smoothed particle hydrodynamics (SPH) models for avalanches are extended in the present work to include a μ(I)- rheological model enriched with a fragmentation law. With this improvement, the basal friction becomes grain distribution dependent. Rock avalanches, where grain distribution tends to change with time while propagating, are the appropriate type of landslide to apply the new numerical proposal. The μ(I)- rheological models considered in the present work are those of Hatano and Gray, com- bined with two different fragmentation laws, a hyperbolic and a fractal‐based law. As an application, Frank avalanche, which took place in Canada in 1903, is analyzed under the scope of the present approach, focusing in the influence of the rheological and fragmentation laws in the evolution of the avalanche. KEYWORDS depth‐integrated models, Frank rock avalanche, inertia number, rheology, rock avalanches, rock fragmentation, SPH 1 | INTRODUCTION: ROCK AVALANCHES Rock avalanches are a particular case of landslides with a very high capacity of destruction. They usually involve large masses of rock that are difficult to stabilize. The mobilized volumes are usually very high. For instance, in the case of Frank avalanche, which will be considered here, the volume was 36 000 000 m 3 . Velocities can also be high, reaching 100 km/h with heights that can be larger than 20 m (five‐storey house). The phenomena leading to rock avalanches are complex, including changes of effective stresses caused by earth- quakes, rain or melting of ice, degradation of material strength due to weathering or chemical attack, or changes of slope geometry caused by human action or erosion. Received: 24 April 2018 Revised: 11 September 2018 Accepted: 24 September 2018 DOI: 10.1002/nag.2912 Int J Numer Anal Methods Geomech. 2019;43:833–857. © 2019 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/nag 833