A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi's effective stress principle

TL;DR

This paper develops a variational framework incorporating second-gradient effects to extend Terzaghi's effective stress principle for saturated porous solids, providing a deeper understanding of deformation behavior.

Contribution

It introduces a second-gradient theory for saturated porous solids that extends classical effective stress principles using variational methods and density-gradient effects.

Findings

Derivation of equilibrium equations with second-gradient effects

Extension of Terzaghi's effective stress principle

Distribution of saturation pressure among constituents

Abstract

The principle of virtual power is used to derive the equilibrium field equations of a porous solid saturated with a fluid, including second density-gradient effects; the intention is the elucidation and extension of the effective stress principle of Terzaghi and Fillunger. In the context of a first density-gradient theory for a saturated solid we interpret the porewater pressure as a Lagrange multiplier in the expression for the deformation energy, assuring that the saturation constraint is verified. We prove that this saturation pressure is distributed among the constituents according their respective volume fraction (Delesse law) only if they are both true density-preserving.