Some axially symmetric flows of Mohr–Coulomb compressible granular materials

Abstract
In this paper we consider a number of axially symmetric flows of compressible granular materials obeying the Coulomb-Mohr yield condition and the associated flow rule. We pay particular attention to those plastic regimes and flows not included in the seminal work of Cox, Eason & Hopkins (1961). For certain plastic regimes, the velocity equations uncouple from the stress equations and the flow is said to be kinematically determined. We present a number of kinematically determined flows and the development given follows the known solutions applicable to the so-called `double-shearing' model of granular materials which assumes incompressibility and for which the governing equations are almost the same. Similarly, for certain other plastic regimes the stresses may be completely determined without reference to the velocity equations and these are referred to as statically determined flows. In the latter sections of the paper we examine statically determined flows arising from the assumption that the shear stress in either cylindrical or spherical polar coordinates is zero. In the final section we present a numerical solution, which incorporates gravitational effects, for the flow of a granular material in a converging hopper. In addition, we examine the Butterfield & Harkness (1972) modification of the double-shearing model of granular materials which formally includes both the double-shearing theory and the Coulomb-Mohr flow rule theory as special cases. Moreover, for kinematically determined regimes, the velocity equations are the same apart from a different constant, while for statically determined regimes the governing velocity equations are slightly more complicated, involving another constant which is a different combination of the basic physical parameters. Thus some of the solutions presented here can be immediately extended to this alternative theory of granular material behaviour and therefore the prospect arises of devising experiments which might validate or otherwise one theory or the other.

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