Keywords: rigid sphere, Biot's theory, harmonic load, pore fluid pressure, dynamic response.

The dynamic behaviour of rigid inclusions represents a useful way to study the dynamic response of foundations embedded in soil media. The dynamic response of rigid spheres subjected to axial harmonic and impulsive loads has an important practical implication in geotechnical engineering, especially with the increasing use of ball penetrometers for in-situ testing of soft deposits [1]. The introduction of full-flow penetrometers over the last few years has led to significant improvement in methods of in-situ soil testing. The (nominally) plane strain T-bar and axisymmetric ball penetrometer are designed to have a projected area about ten times the shaft, which they are attached to, in order to minimise any correction required for the overburden stress. The dynamic response of rigid spheres is also relevant to offshore engineering where different shapes of anchors are often used to support floating structures. The dynamic response of spherical inclusion draws the attention of many researchers. A solution for steady oscillation of a spheroid inclusion in an infinite extent media were previously developed [2,3,4,5]. In this paper, an analytical solution for the response of a rigid sphere embedded in a full space poroelastic medium subjected to a dynamic lateral load is derived. The solution is obtained using Biot's theory for acoustic waves. In this solution, the displacements of the solid skeleton and the pore pressure are expressed in terms of three scalar potentials. These potentials correspond to the wave velocities of the slow and fast compressional waves and to the shear wave. Different boundary and load conditions were investigated. The sphere is taken to be fully-bonded to the surrounding elastic medium. Exact analytical solutions for pore fluid pressure, stresses and displacements in the frequency domain are obtained using the stress function technique. The fast Fourier transform procedure was employed to calculate the inverse Fourier transform. The effect of the permeability and Poisson's ratio of the porous medium on the solution is investigated together with the influence of the mass of the rigid sphere. The results are expressed in non-dimensional forms. The example of harmonic load has demonstrated that Poisson's ratio has little influence on the shape and magnitude of the displacements while the permeability has a significant effect.

1

M.F. Randolph, M.J. Cassidy, S. Gourvenec, C.T. Erbrich, "The Challenges of Offshore Geotechnical Engineering", In: Proceedings of the 16th Int. Conf. on Soil Mechanics and Geotechnical Eng, (ISSMGE), Osaka, Japan, 1, 123-176, 2005.

2

W.E. Williams, "A note on the slow vibrations in an elastic medium", The Quarterly Journal of Mechanics and Applied Mathematics, 19, 413-416, 1966. doi:10.1093/qjmam/19.4.413

3

P. Chadwick, E.A. Trowbridge, "Oscillations of a rigid sphere embedded in an infinite elastic solid, I. Torsional oscillations", Proceedings of the Cambridge Philosophical Society, 63, 1189-1205, 1967.

4

S.K. Datta, "Rectilinear oscillations of a rigid spheroid in an elastic medium", Quartlty Applied Mathematics, 37(1), 86-91, 1979.

5

A.P.S. Selvadurai, G. Dasupta, "Harmonic response of smoothly embedded rigid sphere", ASCE Journal of Engineering Mechanics, 116(9), 1945-1958, 1990. doi:10.1061/(ASCE)0733-9399(1990)116:9(1945)

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