Response of a stringlike dislocation loop to an external stress

Abstract

The dynamics of a very thin dislocation loop under the influence of an externally applied, time dependent, stress field is studied in the context of continuum elasticity, where very thin means that the dislocation core is small compared to the loop's typical radius of curvature as well as to any relevant acoustic wavelengths. This is done using energy and momentum conservation as derived from a variational principle for conservative motion of the loop. Energy conservation alone does not suffice, since it is insensitive to forces that do no work. The idea is to have a theory of sources (dislocation loops) interacting with a field (particle displacement) in the same sense that classical electrodynamics is a theory of point-charged particles interacting with the electromagnetic field. The sum of elastic strain and particle velocity generated by a dislocation loop and those generated by external agents are replaced in the action functional whose extrema give the equations of classical dynamic elasticity, thus obtaining a functional of the loop's trajectory. Extrema of the action with respect to variations of the dislocation history select the trajectory that will be followed by the loop under prescribed external stresses. In general, the evolution of the loop will be governed by an integrodifferential equation. Differential equations are obtained when the work done by external forces is much greater than the elastic energy radiated, and the motion of any one point of the loop is affected only by those other loop points in its immediate neighborhood (local approximation). These equations are explicitly written down. They describe the dynamics of a string with mass and line tension of purely elastic origin. The cutoff procedure needed to give meaning to logarithmically divergent expressions is carefully described. The main ideas can be understood in the case of a screw dislocation, which is worked out in detail. The general case with two characteristic velocities, although algebraically more cumbersome, is not essentially different physically. Additional examples include the gliding edge, pinned dislocation segments, and kinks. Results presented are valid in a homogeneous, isotropic, infinite elastic solid, and ways in which these various restrictions might be lifted is discussed.