Epicenter and epicentral-axis motion of a three-dimensional half-space

Abstract

Ideally, this chapter would be devoted to a three-dimensional treatment of Lamb’s problem for a point load suddenly applied at the surface of a transversely isotropic elastic half-space. For an isotropic solid, the surface displacements for this problem were first successfully treated by Pekeris [48]. In the analysis of Pekeris, the Lamé parameters λ and µ of the solid were assumed to be equal. This assumption permitted an easy solution of the Rayleigh cubic equation, but was not actually crucial to the mathematical technique employed. Pekeris gave a closed form solution for the surface displacements when the applied point loading was normal to the half-space. Although both the normal and tangential components of the surface displacement vector were found explicitly, the tangential component (unlike the normal component) required elliptic integrals of the first and third kinds for its description. The corresponding problem for a transversely isotropic half-space, in which the surface is normal to the axis of material symmetry, has been studied by Ryan [67]. Ryan discusses the wave front shape and singularities in the interior of the solid and gives graphical results for the surface displacements. Unfortunately the three-dimensional version of Lamb’s problem for a transversely isotropic solid does not (in contrast with the two-dimensional problem treated in Chapter 4) admit an explicit solution for surface displacements. Ryan uses numerical integration to evaluate the integral expressions for both components of the surface displacement vector.