  Devised towards geophysical applications for various processes in the
lithosphere or the crust, a model of poro-elastodynamics with inelastic strains
and other internal variables like damage (aging) and porosity as well as with
diffusion of water is formulated fully in the Eulerian setting. Concepts of
gradient of the total strain rate as well as the additive splitting of the
total strain rate are used while eliminating the displacement from the
formulation. It relies on that the elastic strain is small while only the
inelastic and the total strains can be large. The energetics behind this model
is derived and used for analysis as far as the existence of global weak
energy-conserving solutions concerns. By this way, the model in [V. Lyakhovsky
et al., Pure Appl. Geophys., 171:3099-3123, 2014] and [V. Lyakhovsky et al.,
Izvestiya, Physics of the Solid Earth, 43:13-23, 2007] is completed to make it
mechanically consistent and amenable for analysis.
