  A model of saturated hyperelastic porous solids at large strains is
formulated and analysed. The material response is assumed to be of a
viscoelastic Kelvin-Voigt type and inertial effects are considered, too. The
flow of the diffusant is driven by the gradient of the chemical potential and
is coupled to the mechanics via the occurrence of swelling and squeezing.
Buoyancy effects due to the evolving mass density in a gravity field are
covered. Higher-order viscosity is also included, allowing for physically
relevant stored energies and local invertibility of the deformation. The whole
system is formulated in a fully Eulerian form in terms of rates. The energetics
of the model is discussed and the existence and regularity of weak solutions is
proved by a combined regularization-Galerkin approximation argument.
