Effective Stress for Saturated and Unsaturated Porous Media– A Differential Approach

Here we theoretically derive the Terzaghi stress principle for saturated and partially saturated isotropic, porous media with compressible phases. We use previously derived thermodynamic definitions of the drained and unjacketed compressibilities and total differentials to theoretically determine how the total pressure relates to isotropic strain and changes in fluid pressures. We show that under simplifying assumptions we recover the varying forms of the Biot coefficient for saturated porous media and the Bishop parameter for partially saturated porous media. We compare this approach with four modern constructions: the mixture theoretic approaches of Coussy and Borja, Wang's differential approach, and the thermodynamically constrained averaging theory approach of Gray and Schrefler. In doing so we theoretically clarify the often confused definitions of the solid compressibility coefficient, the unjacketed and drained compressibilities, and the generalized Terzaghi stress principle in differential form.