Continuum homogenisation of stochastic comminution with grainsize fabric

Abstract

Comminution is the process of grainsize reduction due to grain crushing, which is common in both natural and industrial problems that involve brittle granular media. Grain crushing is a stochastic process, where the strength of individual grains is determined both by their own size and mineralogy, and the local arrangement of their neighbouring particles, here termed grainsize fabric. The relationship between these effects was described previously using a stochastic lattice model of comminution, whose most general form further includes segregation and mixing. While the stochastic segregation and mixing dynamics have simple analogous differential equations for describing the equivalent continuum behaviour, until now the stochastic comminution dynamics has lacked a smooth representation. Here we resolve this gap by developing a homogeneous continuum model that is based on the same stochastic physics. We show that the new model yields the same results as the stochastic lattice model in the limit of indefinitely large lattices. Given a time varying stress state, the model describes the time-evolution of the grainsize distribution by representing the state of the system in terms of a two-dimensional joint distribution over grainsize and local-average grainsize. The model has a scalable numerical implementation with a simple deterministic rule for time-evolution.