Green’s tensor and its potentials for inhomogeneous elastic media

The explicit exact form of the spectral Green tensor is obtained for 3D inhomogeneous elastic media with a single axis (dimension) of inhomo­geneity. The tensor is specified, in the most general case, by five scalar potentials: axial-dilatational (Φˆz), axial-shear (Ψˆz), transverse-dilatational (Φˆt), transverse-shear (Ψˆt) and horizontal-shear (χˆ), where each potential is effectively a scalar Green function. In general, the two axial potentials are engaged in a pair of coupled inhomogeneous biharmonic wave equations. The transverse pair is coupled via a similar system of equations. The horizontal-shear potential obeys a wave equation of its own. In a restricted class of media, the biharmonic wave equations degenerate into coupled second-order wave equations. In media wherein the dilata­ional and shear wave motions are totally uncoupled (vanishing of the spatially dependent coupling coefficients η12 and η21,) the number of independent potentials reduces further to three (Φ,Ψ,χ) and each potential obeys its own wave equation. Finally, in homogeneous media, the two shear potentials coalesce into a single potential. In the general case, the tensor is asymmetric and its curvilinear components are independent of the azimuth-angle coordinate. The paper exhibits a new mathematical scheme that enables one to calculate the displacement and strain fields in arbitrary 3D inhomogeneous media (with one-dimensional inhomogeneity) due to any given multipolar point source. We treat the displacement fields excited by a single-point force and a general dipolar force system. In particular, we obtain the potential equations for a couple, a centre of rotation (torque) and a displacement dislocation. It is shown, among other things, that a spherically symmetric energy injection into an elastic medium with a non-zero density and rigidity gradients at the source point, will produce a non-symmetric radiation field; the deviation of which from spherical symmetry, may serve to measure these gradients.The potential wave equations are solved in closed form for a constant-gradient solid. Approximation of the potentials under the ‘mode decoupling condi­tion’ is presented and worked in detail for the case of a parabolic velocity profile.