Anatomy of Green's functions for line forces and dislocations in anisotropic media and in degenerate materials

Abstract

It is shown that the Green's function for the infinite space subject to a line force and a line dislocation consists of three independent Green's functions. Each independent Green's function is a one-component Green's function whose displacement u is polarized on the plane spanned by the complex eigenvector a while the surface traction vector tΓ on any boundary Γ is polarized on the plane spanned by the other complex eigenvector b. The dislocation and force required for the one-component Green's function are also on the planes a and b, respectively. For degenerate materials one of the one-component Green's functions has to be modified to include the generalized complex eigenvectors a* and b*. The dislocation and force required for this modified Green's function are on the planes a* and b*, respectively, but the displacement u and the surface traction tΓ are no longer polarized on a single plane. The corresponding problems for half-spaces are also investigated. There are again three independent Green's functions for half-spaces for general anisotropic materials and for degenerate materials. Finally we present alternate expressions of Green's functions for degenerate materials which are more useful in practical applications.