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Variational formulation of pre-stressed solid-fluid mixture theory, with an application to wave phenomena. (English) Zbl 1146.74012
Summary: Fluid-saturated porous media are modelled by the theory of mixtures, and the placement maps of solid and 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 external forces and the variation of such an action along each variation are zero. Constitutive equations for two Cauchy stress tensors and for the interaction force are derived taking into account a general state of pre-stress for solid and fluid species. Governing equations are formulated, however, for the sake of simplicity, only in the case of pure initial pressure. 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.

MSC:
74F20 Mixture effects in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74J10 Bulk waves in solid mechanics
74H55 Stability of dynamical problems in solid mechanics
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