Non-crystallographic motion of dislocations as a fine mixture of rectlinear paths

Paolo Cermelli
Dipartimento di Matematica

We discuss the convergence of an approximation scheme for the solution, near an attractor, of a discontinuous dynamical system arising in the theory of dislocations in crystalline solids. It is well known that dislocations can only move along a finite number of crystallographic directions: in two dimensions, the resulting trajectories are piecewise rectilinear paths. However, in special situations such as near an attractor, dislocations are forced to move along curved paths: we characterize this class of motions as fine mixtures of crystallographic motions, using the notion of generalized curves due to L. C. Young, and explicitly compute the parametrized measure associated to a sequence of polygonals. The result is then used to motivate a simple numerical scheme, and show that it is physically consistent.