Evaluation of the anisotropic Green's function in three dimensional elasticity

Abstract

A perturbation expansion technique for approximating the three dimensional anisotropic elastic Green's function is presented. The method employs the usual series for the matrix (I−A)^-1 to obtain an expansion in which the zeroth order term is an isotropic fundamental solution. The higher order contributions are expressed as contour integrals of matrix products, and can be directly evaluated with a symbolic manipulation program. A convergence condition is established for cubic crystals, and it is shown that convergence is enhanced by employing Voigt averaged isotropic constants to define the expansion point. Example calculations demonstrate that, for moderately anisotropic materials, employing the first few terms in the series provides an accurate solution and a fast computational algorithm. However, for strongly anisotropic solids, this approach will most likely not be competitive with the Wilson-Cruse interpolation algorithm.