Elastic Green's Function for Anisotropic Cubic Crystals

Abstract

The Green's function describing the elastic displacement due to a unit force in an infinite cubic material is investigated in detail. Only for special cases can an exact solution be given, i.e., for $c_{11}-c_{12}-2\mathrm{}{c}_{44}=0\mathrm{}$ (isotropy), for $c_{12}+c_{44}=0\mathrm{}$, and for directions. Perturbation theory is applied to the cases where these conditions are only approximately fulfilled. Divergencies or strong maxima of the Greens' function, occurring in nearly unstable materials for $c_{11}-c_{12}\to 0$ or $c_{44}\to 0$, are examined. Analytical approximations for the Green's function are given by fitting the exact known Fourier transform with a suitably chosen ansatz in certain directions. Other simple approximations are derived by variational techniques and give good results for crystals with small and medium anisotropy.