Keywords: wave propagation, porous media, Biot theory, surface vibration, Fourier transform, wave number.

The study of mechanical wave propagation in soils has received considerable attention in the last decades. As a matter of fact, a better understanding of propagation phenomena would allow a better apprehension of problems in a large number of areas of applied mechanics or geomechanics. In the 1990s, Jones et al. [1] developed a semi-analytical approach to analyse displacements in the wave number and spatial domains for an harmonic load acting on a viscoelastic ground. Extensions of this work have been done for the study of moving loads, Lefeuve-Mesgouez et al. [2] and Sheng et al. [3]. All this research was restricted to the case of viscoelastic soils. Degrande et al. [4] also developed a similar approach for porous media.

The authors propose to develop such an approach in the case of porous media using the whole Biot theory [5], in order to analyse results obtained in the wave number domain and to understand dispersion and attenuation phenomena. In fact, the study of the displacements in the wave number domain allows localization and dissociation of body and surface waves and this can be useful in the interpretation of results in the spatial domain in which all waves are mixed.

An harmonic vertical load acts uniformly over a rectangle. It rests on an homogeneous porous soil which is modelled as a two-phase continuum composed of a porous deformable viscoelastic solid skeleton and a fluid component corresponding to the viscous fluid which saturates the porous space. The model is three-dimensional but it can be easily restricted to two-dimensional geometries. The equations field is nondimensionalised to present the problem in a general and meaningful manner. By introducing the Helmholtz decomposition for the solid and relative displacements in the equations of motion and in the stress-strain relations, one can obtain the wave equations relative to the problem that underline the existence of the body waves. As a matter of fact, in such a medium, three body waves exist: the and compressional waves and the shear wave. Moreover, surface waves such as the Rayleigh wave also exist. A Fourier transform on the surface spatial variables is used to solve the problem in the transformed domain. A differential system over the third spatial variable is obtained and solved with appropriate boundary conditions: this yields the "transformed" displacements or the displacements in the wave number domain. The inverse Fourier transform is then performed numerically to obtain the displacements in the spatial domain.

Results presented in this paper are obtained for the case of a two-dimensional problem involving a strip load over the surface of the half-space. The analysis of the real part of the transformed displacements dissociates the contribution of the different waves: contribution of the body waves have opposite signs compared to the contribution of the surface wave. The major features are peaks located at corresponding wave numbers. The wave, specific to porous media, can be clearly visualized on the fluid phase even if its contribution is lower than the other body waves. To the authors' knowledge, such an analysis has not been carried out yet.

The influence of the permeability has also been studied. It has important consequences on the wave contribution and on the monophasic or biphasic behavior of the ground. Decreasing permeability makes the coupling between solid and fluid phases higher: for instance, for low values of permeability, the wave appears on the fluid phase because of the coupling. The influence of the wave is more important for higher permeability. The solid phase displacement has the same appearance for each permeability value. Thus, permeability has greater influence on the fluid phase: when permeability is high, fluid tends to flow out rather than to deform, which explains lower amplitude of displacements. This result is in agreement with other results presented by the authors in the case of transient regimes and obtained with a specific finite element code developed at the laboratory.

The main perspective of this work is to deal with transient regimes and multilayered soils. Such semi-analytical approaches can also been used as benchmarks to validate numerical tools.

1

D.V. Jones, D. Le Houédec, M. Petyt, "Ground vibration due to a rectangular harmonic load", Journal of Sound and Vibration, 212(1), 61-74, 1998. doi:10.1006/jsvi.1997.1367

2

G. Lefeuve-Mesgouez, A. T. Peplow, D. Le Houédec, "Vibration in the vicinity of a high-speed moving harmonic strip load", Journal of Sound and Vibration, 231(5), 1289-1309, 2000. doi:10.1006/jsvi.1999.2731

3

X. Sheng, C.J.C. Jones, M. Petyt, "Ground vibration generated by a harmonic load acting on a railway track", Journal of Sound and Vibration, 225(1), 3-28, 1999. doi:10.1006/jsvi.1999.2232

4

G. Degrande, G. De Roeck, P. Van Den Broeck, D. Smeulders, "Wave propagation in layered dry, saturated and unsaturated poroelastic media", International Journal of Solids and Structures, 35, 4753-4778, 1998. doi:10.1016/S0020-7683(98)00093-6

5

T. Bourbié, O. Coussy, R. Zinszner, "Acoustique des milieux poreux", Technip, Paris, 1986.

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