  In this paper, we study non-isochoric models for mixtures of solid particles,
at high volume concentration, and a gas. One of the motivations of this work
concerns geophysics and more particularly the pyroclastic density currents
which are precisely mixtures of pyroclast and lithic fragments and air. They
are extremely destructive phenomena, capable of devastating urbanised areas,
and are known to propagate over long distances, even over almost flat
topography. Fluidisation of these dense granular flows by pore gas pressure is
one response that could explain this behaviour and must therefore be taken into
account in the models. Starting from a gas-solid mixing model and invoking the
compressibility of the gas, through a law of state, we rewrite the conservation
of mass equation of the gas phase into an equation on the pore gas pressure
whose net effect is to reduce the friction between the particles. The momentum
equation of the solid phase is completed by generic constitutive laws,
specified as in Schaeffer et al (2019, Journal of Fluid Mechanics, 874,
926-951) by a yield function and a dilatancy function. Therefore, the
divergence of the velocity field, which reflects the ability of the granular
flow to expand or compress, depends on the volume fraction, pressure, strain
rate and inertial number. In addition, we require the dilatancy function to
describe the rate of volume change of the granular material near an isochoric
equilibrium state, i.e. at constant volume. This property ensures that the
volume fraction, which is the solution to the conservation of mass equation, is
positive and finite at all times. We also require that the non-isochoric
fluidised model is linearly stable and dissipates energy (over time). In this
theoretical framework, we derive the dilatancy models corresponding to
classical rheologies such as Drucker-Prager and $\mu$(I) (with or without
expansion effects). The main result of this work is to show that it is possible
to obtain non-isochoric and fluidised granular models satisfying all the
properties necessary to correctly account for the physics of granular flows and
being well-posed, at least linearly stable.
