Reconciliation of the Effective Stress Equation between Macroscopic and Microscopic Static Equilibria

The effective stress equation is considered as the cornerstone of soil mechanics, yet it has been subject to diverse interpretations and debates since its inception. Efforts have been made to study the effective stress equation at the microscopic level for a more refined understanding. However, the well-known microscopic effective stress equations have not appeared compatible with the macroscopic effective stress equations, leading to confusion. To address this issue, a reconciliation of the effective stress equation at both macroscopic and microscopic levels is proposed. The generalized macroscopic effective stress equation is derived from static equilibrium on any cross section of a representative elementary volume of soil in the normal direction, with consideration of skeleton water. In parallel, the general static equilibrium involving local active stresses at the microscopic level is presented and compared with the macroscopic effective stress equation. An equation is then proposed to relate the local active stresses at the microscopic level to an equivalent stress transmitted through skeleton water at the macroscopic level. This equation reconciles the macroscopic and microscopic effective stress equations in general, as well as the parallel and series models for saturated soils in particular. Furthermore, skeleton water is linked to the soil–water characteristic curve, enabling its approximate determination. The proposed approach offers a new perspective for understanding effective stress at both the microscopic and macroscopic levels.