Elastic Green's Function for Anisotropic Cubic Crystals

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 ${c}_{11}\ensuremath{-}{c}_{12}\ensuremath{-}2{c}_{44}=0$ (isotropy), for ${c}_{12}+{c}_{44}=0$, and for 100> 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 ${c}_{11}\ensuremath{-}{c}_{12}\ensuremath{\rightarrow}0$ or ${c}_{44}\ensuremath{\rightarrow}0$, 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.