Hydrodynamics of stresses and pressures in saturated and unsaturated deformable porous media

Stresses and pressures are used to represent the hydromechanical state of deformable porous media. Past formulations often adopt the effective stress principle, usually in an empirical and energetically inconsistent way. Using the rigorous hydrodynamic procedure, this study pursues an alternative energy-consistent formulation for the full characterisation of both saturated and unsaturated porous materials. An elastic stress is consistently linked to its energy-conjugated elastic strain and, in the absence of viscous stress, has a structure that was previously interpreted as an effective stress. Here, it is emphasised that this similarity does not imply that the elastic stress is ‘effective’ in the classical sense, namely that it can replace total stress in dry soils to represent the mechanical behaviour of saturated or unsaturated soils. The dependence of the elastic stress on the deformability of the solid is incorporated constitutively using a general elastic strain energy of pressure- and density-dependent media, excluding energy costs from solid density changes due to volumetric elastic straining. By adopting the resulting internal energy that is convex for physically realistic porous materials, the proposed formulation yields a rigorous quantification of the elastic stress, and the pressures of the air, water, and solid required for characterising saturated and unsaturated soils, including the Biot stress correction coefficient for deformable porous media at variable saturation. The formulation also reveals the intrinsic dependence of the stress coefficients on material elasticity and the characteristics of water retention responses.