One of the basic solutions for anisotropic elasticity, and for other subject for that matter, is the Green's function. The importance of Green's functions in constructing solutions to boundary value problems has been well recognized. We will study in this chapter Green's functions for infinite spaces, half-spaces, and bimaterials that consist of two half-spaces of dissimilar materials bonded together. Also studied are a composite space that consists of wedges of dissimilar materials and an angularly inhomogeneous space. Green's functions for the infinite space with the presence of a crack, an elliptic hole, or an elliptic inclusion will be studied in separate chapters. We will be concerned mainly with Green's functions due to a line of concentrated forces and a line dislocation that have the r-1 stress singularity where r is the radial distance from the line of forces or the line dislocation. Green's functions due to a concentrated couple, a double force, a center of dilatation, etc., that provide the r-2 stress singularities will be discussed in Section 8.12. We will see that most solutions can be expressed in a real form with the identities presented in Chapters 6 and 7. The Green's function for two-dimensional deformations of an infinite anisotropic elastic material subject to a line dislocation has been obtained by Eshelby et al. (1953), Stroh (1958), Willis (1970), Malen and Lothe (1970), and Malen (1971). Further developments of the Green's function to include a line force was given by Barnett and Lothe (1975a). The solution was in a complex form. A real form solution using an integral representation was derived by Barnett and Swanger (1971) and Asaro et al. (1973) (see also Mura, 1975). Most of the real form solutions obtained by these authors were for the displacement gradient, and hence for the strain. The stress was then obtained indirectly through the stress-strain relations. Chadwick and Smith (1977) did present real form solutions for the displacement as well as the stress directly. The solutions required the inverse of the 6x6 matrix (x1I + x2N) which can be achieved by employing (7.9-17).
Fast Multipole Representation of Green's Function for an Impedance Half-Space