The Double-Shearing Velocity Equations for Dilatant Shear-Index Granular Materials

Abstract

Modelling the flow of a granular material constitutes one of the outstanding problems of continuum mechanics, for which there is little general agreement on the governing mathematical equations. This is because real granular materials are not readily defined and have widely varying physical characteristics, which give rise to behaviour which is so complex and variable that a single mathematical model is unlikely to account for the behaviour of all granular materials under all practical or experimental conditions. However, it is generally accepted that failure of a granular material occurs due to frictional slip between particles and that the material yields at a surface when the magnitude of the shear component of stress τ assumes a particular relationship with the normal component of stress σ, namely |τ| = f(σ) for some function f(σ), where here we take the normal component of stress to be positive in tension. A large body of experimental work supports this hypothesis and it would be true to say that the resulting stress distribution, based on this assumption, is reasonably in accord with experimental evidence and generally accepted as an overall accurate model. In contrast there is very little general agreement on the associated velocity or displacement field corresponding to a given stress distribution and the topic is controversial.