To model cohesionless granular flow using continuum theory, the usual approach is to assume the cohesionless Coulomb-Mohr yield condition. However, this yield condition assumes that the angle of internal friction is constant, when according to experimental evidence for most powders the angle of internal friction is not constant along the yield locus, but decreases for decreasing normal stress component σ from a maximum value of π/2. For this reason, we consider here the more general yield function which applies for shear-index granular materials, where the angle of internal friction varies with σ. In this case, failure due to frictional slip between particles occurs when the shear and normal components of stress τ and σ satisfy the so-called Warren Spring equation (|τ|/c)n = 1 − (σ/t), where c, t and n are positive constants which are referred to as the cohesion, tensile strength and shear-index respectively, and experimental evidence indicates for many materials that the value of the shear-index n lies between 1 and 2. For many materials, the cohesion is close to zero and therefore the notion of a cohesionless shear-index granular material arises. For such materials, a continuum theory applying for shear-index cohesionless granular materials is physically plausible as a limiting ideal theory, and any analytical solutions might provide important benchmarks for numerical schemes. Here, we examine the cohesionless shearindex theory for the problem of gravity flow of granular materials through two-dimensional wedge-shaped hoppers, and we attempt to determine analytical solutions. Although some analytical solutions are found, these do not correspond to the actual hopper problem, but may serve as benchmarks for purely numerical schemes. The special analytical solutions obtained are illustrated graphically, assuming only a symmetrical stress distribution.
Cox, G. M. & Hill, J. M. (2004). The limiting ideal theory for shear-index cohesionless granular materials. ANZIAM Journal, 45 (3), 373-392.