Green’s tensor and its potentials for inhomogeneous elastic media

Abstract

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 (Φ ˆ z ), axial-shear (Ψ ˆ z ), transverse-dilatational (Φ ˆ t ), transverse-shear (Ψ ˆ t ) and horizontal-shear (χ ˆ ), 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 η 12 and η 21 ,) 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.