Anisotropic elasticity with applications to dislocation theory

Abstract

The general solution of the elastic equations for an arbitrary homogeneous anisotropic solid is found for the case where the elastic state is independent of one (say x 3) of the three Cartesian coordinates x 1, x 2, x 3. Three complex variables z l = x 1 + p l x 2 ( l = 1, 2, 3) are introduced, the p l being complex parameters determined by the elastic constants. The components of the displacement ( u 1, u 2, u 3) can be expressed as linear combinations of three analytic functions, one of z ( l) , one of z (2), and one of z (3). The particular form of solution which gives a dislocation along the x 3-axis with arbitrary Burgers vector ( a 1, a 2, a 3) is found. (The solution for a uniform distribution of body force along the x 3-axis appears as a by-product.) As is well known, for isotropy we have u 3= 0 for an edge dislocation and u 1 = 0, u 2 = 0 for a screw dislocation. This is not true in the anisotropic case unless the x 1 x 2 plane is a plane of symmetry. Two cases are discussed in detail, a screw dislocation running perpendicular to a symmetry plane of an otherwise arbitrary crystal, and an edge dislocation running parallel to a fourfold axis of a cubic crystal.