Finite element modelling of saturated porous media at finite strains under dynamic conditions with compressible constituents

Two finite element formulations are proposed to analyse the dynamic conditions of saturated porous media at large strains with compressible solid and fluid constituents. Unlike similar works published in the literature, the proposed formulations are based on a recently proposed hyperelastic framework in which the compressibility of the solid and fluid constituents is fully taken into account when geometrical non-linear effects are relevant on both micro- and macroscales. The first formulation leads to a three-field finite element method (FEM), which is suitable for analysing high-frequency dynamic problems, whereas the second is a simplification of the first, leading to a two-field FEM, in which some inertial effects of the pore fluid are disregarded, hence the second formulation is suitable for studying low-frequency problems. A fully Lagrangian approach is considered, hence all terms are expressed with reference to the material setting; the balance equations for the pore fluid are also expressed in terms of the chemical potential and the mass flux of the pore fluid in order to take the compressibility of the fluid into account. To improve the numerical response in the case of wave propagation, a discontinuous Galerkin FEM in the time domain is applied to the three-field formulation. The results are compared with analytical and semi-analytical solutions, highlighting the different effects of the discontinuous Galerkin method on the longitudinal waves of the first and second kind.