Finite strain hyperelastoplastic modelling of saturated porous media with compressible constituents

Starting from the general expression of the free-energy function of a saturated porous medium at finite deformations in the case of compressible fluid and solid constituents, and from the internal dissipation increment, the general expressions of the plastic potential and flow rule are deduced together with the general form of the consistency condition. Reference is made to an elementary volume moving with the solid skeleton in a Lagrangian description, which is treated as an open system from which the pore fluid can flow freely in and out. As a result, a generalisation is provided of the classical Prandtl–Reuss relationship of small strain elastoplasticity in single-phase media to finite strain multiplicative (for F) and additive (for the fluid mass content) elastoplasticity in saturated porous media with compressible constituents. The following particular cases are analysed in detail: null plastic volume change of the solid constituent, incompressibility of the solid constituent, incompressibility of both fluid and solid constituents, quasi-linear theory (in which the solid constituent is assumed to be nearly incompressible, and therefore undergoing small volume changes), and geometrically linearised theory. The simplified approaches previously presented in the literature are thus recovered within a unified framework and new, simplified constitutive assumptions are made