An exact closed-form solution of three-dimensional transient Green's functions in a constrained transversely isotropic half-space

Abstract

In this paper, we derive an exact closed-form solution of three-dimensional transient Green's functions (GFs) at the free surface of a constrained transversely isotropic (TI) half-space due to an arbitrarily oriented surface point force of Heaviside time variation. The solution clearly delineates the distinct features of body waves (P-, SV- and SH- waves) as well as Rayleigh surface wave in displacement components and demonstrates the travel time of each wave amplitude in terms of inequalities among elastic coefficients. It is further shown that owing to material anisotropy, the radial and angular displacements due to horizontal point force may contain a special disturbance, which attenuates proportionally to the inverse square of epicentral distance from the source whilst varies linearly with time. This novel feature is attributed to the peculiarities of shear wave splitting, a consequence of polarization, with different velocities, of the SV- and SH- waves in TI materials, and as such, does not exist in isotropic materials. It is also worthwhile to mention that for this constrained anisotropy, the Rayleigh wave function is degenerated into a new elegant formulation whose real root specifies the Rayleigh wave velocity in terms of elastic coefficients. Moreover, the transient GFs are computed for the entire range of the elastic coefficients including also the case when the Rayleigh function has complex conjugate roots. For validation, the new GFs derived in this paper are suitably degenerated into the corresponding solutions for the isotropic (Poisson's) material and are compared with the existing solutions. The effect of Poisson's ratio on the isotropic response functions is graphically illustrated and so is the effect of material anisotropy for different synthetic TI materials.