Accurate modeling of snow avalanches is indispensable for mapping and mitigating hazards in mountainous areas. While current models are mostly based on depth-averaged flow equations with simple viscoplastic laws, particle-based or mesh-free continuum schemes have become increasingly popular in simulating full three-dimensional granular avalanches as they can readily handle large deformation problems. These include the particle finite element method (PFEM), smoothed particle hydrodynamics (SPH) and the material point method (MPM) which do not suffer from mesh-distortion issues and offer trivial treatment of the free-surface. Here, we present an elastic-viscoplastic constitutive model based on the combination of the µ(I)-rheology and critical state soil mechanics, implemented in a B-spline MPM scheme. With the µ(I)-rheology originally developed for dense granular flows and critical state soil mechanics developed for granular solids, this model offers the “best of both worlds” in the appropriate flow rate limits. Compellingly, it captures cohesive flows and facilitates the treatment of dilatancy and compressibility through the hardening law of a Modified Cam-Clay yield criterion. We demonstrate and verify our model on various setups and problems, including collapse and flow on inclined planes, flow over obstacles and finally full-scale avalanche simulations.