On the traction boundary value problem in the linearized gauge theory of dislocations

The static traction boundary value problem for finite material bodies is shown to be well posed in the linearized gauge theory of dislocations. The dislocation field variables assume the roles of generalized stress potentials that satisfy a system of fourth order linear partial differential equations. Accordingly, the stress distributions may be calculated directly from the traction boundary data without solving for the elastic displacement fields. Satisfaction of appropriate gauge conditions are shown to lead to significant simplifications and certain systems of first integrals of the governing equations are exhibited. The important thing here is that the gauge theory of dislocations provides direct means of calculating the distributions of dislocations that arise from given systems of boundary tractions. This is in sharp contrast with previous theories in which the distributions of dislocations are calculated from given distributions of dislocation densities.