Hydrodynamic derivation of the work input to fully and partially saturated soils

In 'Rational Mechanics' the work input per unit element volume can be identified from the local energy balance, itself being derived through localization from the global energy balance of an arbitrary volume. On the other hand, in the search of meaning for an effective stress of partially and fully saturated soil media, in 'Soil Mechanics' some define the work input by adopting material assumptions and simplified homogenization. The final result depends on effective stress and various pressures that are not seen in the rational mechanics work input expression. To resolve the discrepancy in the structure of the work input, this paper establishes a third approach based on the 'Hydrodynamic procedure', which operates solely at the level of unit volumes and thus does not require localization. The derived equations fully agree with Rational Mechanics, and yield the same work input expressions. The bonus for employing the more comprehensive Hydrodynamic procedure is that it also reveals the meaning of the effective stress in relation to the various pressures.