Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

Fluid saturated porous media are modelled by the theory of mixtures and the placement maps of the solid and of the fluid are considered. The momentum balance equations are derived in the framework of a variational approach: We take an action functional and two families of variations and assume that the sum of the virtual work of the external forces and the variation of such an action along each variation are zero. Constitutive equations for the two Cauchy stress tensors and for the interaction force are derived taking into account a general state of pre-stress for the solid and for the fluid species. Governing equations are therefore formulated, however, for the sake of simplicity, only the case of pure initial pressure is further investigated. The propagation of bulk (transversal and longitudinal) waves and the influence of pre-stress are studied: In particular, stability analyses are carried out starting from dispersion relations and the role of pre-stress is investigated. Finally, a numerical example is established for a given state of pre-stress, deriving the phase velocities and the attenuation coefficients of transversal and longitudinal waves.