Time-harmonic problem for a non-homogeneous half-space with exponentially varying shear modulus

Previously published analytical solutions of dynamic problems for continuously nonhomogeneous bases concerned one-dimensional, plane or axisymmetric cases. In this paper a solution in cylindrical co-ordinates is presented for an arbitrary angle distribution in the horizontal plane. The medium is assumed as isotropic, continuously non-homogeneous in the depth direction and homogeneous in the horizontal direction. Poissonʹs ratio is adopted as constant. For each angle component of the solution including cos(ng) or sin(n~), the problem is reduced to three ordinary differential equations (or two for the axisymmetric case where n = 0) ; two of them are coupled. Corresponding boundary conditions are formulated for given stresses or displacements at planes z = const. An example of non-homogeneity where shear modulus increases exponentially with depth, G(z) = G(O) exp(z/zo), is considered (z0 is a constant). The solution for the half-space subjected to a surface load is represented in the form of integrals including Bessel functions and suitable solutions of above-mentioned ordinary differential equations. At low frequencies the integrands have no singularities on the real axis of the complex plane ; then, beginning from a definite value of the frequency (cutoff frequency), poles of integrands appear on the real axis and energy can be passed to the half-space. At some frequencies (resonance frequencies) there are double poles on the real axis leading to infinite amplitudes in the non-dissipative case. For calculations, shear modulus was treated as a complex quantity (G(0) = G0(1 +ie)), where e is a small positive constant. Results of calculations for surface displacements induced by vertical and horizontal acting point forces on the surface of the half-space are presented for static and dynamic problems, and comparison with results for the homogeneous half-space is demonstrated. © 1997 Elsevier Science Ltd.