On the mechanical energy and effective stress in saturated and unsaturated porous continua

Abstract

This paper has two objectives: (a) to formulate a mechanical theory of porous continua within the framework of strong discontinuity concept commonly employed for the analysis of strain localization in inelastic solids; and (b) to introduce an effective stress tensor for three-phase (solid–water–air) partially saturated porous continua emerging from principles of thermodynamics. To achieve the first objective, strong forms of the boundary-value problems are compared between two formulations, the first in which the velocity jump at the solid–fluid interface is treated as a strong discontinuity problem, and the second in which the strong discontinuity is smeared in the representative volume element. As for the second objective, an effective Cauchy stress tensor of the form σ ¯ = σ + ( 1 - K / K s ) p ¯ 1 emerges from the formulation, where σ is the total stress tensor, K and K s are the bulk moduli of the solid matrix and solid phase, respectively, and p ¯ is the mean pore water and pore air pressures weighted according to the degree of saturation. We show from the first and second laws of thermodynamics that this effective stress tensor is power-conjugate to the solid rate of deformation tensor, and that it includes the mechanical power required to compress the solid phase.