A theory intended for slow, dense flows of cohesionless granular
materials is developed
for the case of planar deformations. By considering granular flows on very
scales, one can conveniently split the individual particle velocities into
mean transport components, and employ the notion of granular temperature
plays a central role in rapid granular flows. On somewhat larger scales,
one can think
of analogous fluctuations in strain rates. Both kinds of fluctuations are
utilized in the
present paper. Following the standard continuum approach, the conservation
for mass, momentum and particle translational fluctuation energy are presented.
The latter two equations involve constitutive coefficients, whose determination
of the main concerns of the present paper. We begin with an associated
flow rule for
the case of a compressible, frictional, plastic continuum. The functional
of the flow rule is chosen so that the limiting behaviours of the resulting
relations are consistent with the results of the kinetic theories developed
flow regimes. Following Hibler (1977) and assuming that there are fluctuations
strain rates that have, for example, a Gaussian distribution function,
it is possible
to obtain a relationship between the mean stress and the mean strain rate.
out, perhaps surprisingly, that this relationship has a viscous-like character.
shear rates, the constitutive behaviour is similar to that of a liquid
in the sense that
the effective viscosity decreases with increasing granular temperature,
rapid granular flows, the viscosity increases with increasing granular
in a gas. The rate of energy dissipation can be determined in a manner
that used to derive the viscosity coefficients. After assuming that the
the strain-rate fluctuations can be related to the granular temperature,
a closed system of equations that can be used to solve boundary value problems.
The theory is used to consider the case of a simple shear flow. The resulting
for the stress components are similar to models previously proposed on
more ad hoc basis in which quasi-static stress contributions were
directly patched to
rate-dependent stresses. The problem of slow granular flow in rough-walled
chutes is then considered and the velocity, concentration and granular
profiles are determined. Thin boundary layers next to the vertical sidewalls
the concentration boundary layer being thicker than the velocity boundary
This kind of behaviour is observed in both laboratory experiments and in
dynamics simulations of vertical chute flows.