Generalized Terzaghi's Effective Stress Equation for Unsaturated Soil: An Independent Phase Balance Approach That Considers a Pore Water Content Gradient

The effective stress equation for unsaturated soil is the most important equation in unsaturated soil mechanics. It has been derived by many scholars using different methods. However, none of them considered the gradient of the pore water content, which results in unreasonable force balance equations for different constituent phases in unsaturated soil. To introduce the gradient, we propose an extended three-phase physical model that includes capillary water, air, and generalized soil skeletons. Based on this model, three balance equations for these three constituent phases are separately formulated by considering the gradient of the pore water content. Comparing the result of the superposition of these three balance equations with the total balance equation, we derive a generalized Terzaghi's effective stress equation. This equation states that the effective stress is equal to the total stress minus the neutral stress. In comparison with the classical Bishop's equation, the generalized Terzaghi's equation ensures a smooth and continuous transition from unsaturated to saturated conditions not only in mathematical expression but also in physical meaning. Furthermore, the different pressure effects of capillary water and adsorbed water, their volumetric (or areal) effects, and the transformation between them can be considered by adopting the effective saturation of the capillary water as the effective stress parameter. Therefore, the generalized Terzaghi's equation can provide a better choice for estimating the effective stress in unsaturated soils.