Dynamic dislocation pile-ups

Abstract
The motion of a continuous distribution of dislocations moving under a constant imposed stress towards a small locked dislocation is considered. This problem can be modelled as a non-linear partial differential equation for a complex stress function if a linear stress–velocity law is assumed. If the initial density of the distribution is sufficiently small. the dynamic pile-up that occurs can be described using matched asymptotic expansions. In particular, the force on the locked dislocation is derived as a function of time, It is further shown that the solution remains valid when the velocity in proportional to any positive power of the stress.