Global solutions to problems in the classic theory of static dislocations

Abstract

Problems in the classic theory of static dislocations are shown to be well posed and solvable whenever an associated elasticity problem is well posed and solvable. The presence of nonzero distributions of dislocation density show up in distributions of body forces for the associated elasticity problem. The body forces for the associated elasticity problem are explicitly computed in terms of the dislocation distributions. If the assigned distributions of dislocation density have compact support, the anelastic parts of the stress and strain tensors are shown to decay like r −1 in regions where the dislocation densities all vanish. The straight screw dislocation problem is solved explicitly. The solution agrees exactly with the classical solution outside the dislocation core. Since the total stress, strain, and elastic displacement fields are also explicitly determined inside the dislocation core in terms of the assigned distribution of dislocation densities, the solution is global. Although the total elastic displacements are shown to vanish everywhere, evaluation of the distortion outside the dislocation core leads to effective anelastic displacement functions that agree with the Volterra solution. All physically relevant quantities are determined globally (both inside and outside the dislocation cores), while the effective anelastic displacement functions only exist in regions where all of the dislocation densities vanish.