Three-dimensional electrostatic Green's function for general anisotropic linear elastic solids

Abstract

A new solution is obtained for the Green's function for a three-dimensional space of general anisotropic elastic medium subject to a unit point force at x = 0. The novel features of the new solution are that (i) it is explicit in terms of the Stroh eigenvalues pv (v = 1, 2, 3) on the oblique plane whose normal is the position vector x, and (ii) it remains valid for the degenerate cases p1 = p2 and p1 = p2 = p3. The classical solution by Lifshitz and Rozenzweig has the feature (i) but not (ii). Moreover their solution is less explicit than the solution presented here. Other explicit solutions in the literature may have the feature (i) but they require computation of the Stroh eigenvectors. The Stroh eigenvalues pv are the roots with positive imaginary part of a sextic algebraic equation. The Green's function is particularly simple when the sextic equation is a cubic equation in p2. This is the case for any point in a transversely isotropic material and for points on a symmetry plane of cubic materials and monoclinic materials. Application to these materials yields new results that are either unavailable in the literature, or simpler than those available in the literature. We also present Green's function for points on the normal to the symmetry plane of monoclinic materials.