A mathematical and numerical model for finite elastoplastic deformations in fluid saturated porous media

Finite elastic or elastoplastic strains in fluid saturated soils are studied. Isothermal and quasi-static loading conditions are considered. The governing equations at the macroscopic level are derived in a spatial and a material setting. The constituents are assumed to be materially incompressible at the microscopic level. The elasto-plastic behaviour of the solid skeleton is described by the multiplicative decomposition of the deformation gradient into an elastic and a plastic part; the elasto-plastic evolution laws are developed in the spatial setting. The Kirchhoff effective stress tensor and logarithmic principal strains are used in conjunction with an hyperelastic free energy function. The effective stress state is limited by the von Mises or the Drucker-Prager yield surface with isotropic hardening. Algorithmically, a particular “apex formulation” is advocated for the latter case. The fluid is assumed to obey Darcy’s law. The consistent linearisation of the fully non-linear coupled system of equations is derived. A spatial finite element formulation is presented. Numerical examples highlight the developments.