Time-harmonic problem for a nonhomogeneous 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. Poisson's ratio is adopted as constant. For each angle component of the solution including cos(n theta) or sin(n theta), 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(0) exp(z/z(0)), is considered (z(0) 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) = G(0)(1 + i epsilon)), where epsilon 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. (C) 1997 Elsevier Science Ltd.