On the effective stress for unsaturated soils with residual water

In the present work an expression describing the rate of internal energy density for unsaturated soils is derived. The main difference here, with respect to preceding works, lies in the fact that the free and residual water are treated as two different phases from the soil. The latter presents intrinsic velocity equal to that of the solid phase, and is able to exchange mass with the free water phase. Under these conditions, a Bishop-type effective stress is identified as the energy-conjugated variable with the solid deformation rate, and presents a Bishop parameter equal to the effective degree of saturation. Furthermore, the negative rate of effective degree of saturation is energy-conjugated with the suction times the effective porosity. Other contributors to the internal energy come from the constituent compressibilities, the relative flow and the mass exchange between the free and residual water.