Some aspects of flow in granular materials in hoppers

The main part of the paper is devoted to the description of the localized deformation zones in plane (slow) flow of granular materials from hoppers. Specifically, two types of zones are discussed in detail: material layers, where the particles undergo deformation without crossing the band, and shocks, where the deformation takes place when the particles cross the band. The fundamental requirements and assumptions underlying the description of the two types of localized deformation are analysed. Examples of quasi–static solutions within the framework of plasticity theories are presented, which demonstrate the possibility of predicting the type of hopper flow (mass/funnel), and the periodic occurrence of localized deformation zones. Also, the necessary conditions for a stationary shock taking into account the inertia of particles crossing the band are derived. It is demonstrated that some hardening is required for a stationary shock to exist. Lastly, the relevance of an appropriate constitutive model of granular material for describing the formation of arches and empty channels which obstruct continuous gravitational flow is addressed.

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