  Conventional discrete-to-continuum approaches have seen their limitation in
describing the collective behaviour of the multi-polar configurations of
dislocations, which are widely observed in crystalline materials. The reason is
that dislocation dipoles, which play an important role in determining the
mechanical properties of crystals, often get smeared out when traditional
homogenisation methods are applied. To address such difficulties, the
collective behaviour of a row of dislocation dipoles is studied by using
matched asymptotic techniques. The discrete-to-continuum transition is
facilitated by introducing two field variables respectively describing the
dislocation pair density potential and the dislocation pair width. It is found
that the dislocation pair width evolves much faster than the pair density. Such
hierarchy in evolution time scales enables us to describe the dislocation
dynamics at the coarse-grained level by an evolution equation for the slowly
varying variable (the pair density) coupled with an equilibrium equation for
the fast varying variable (the pair width). The time-scale separation method
adopted here paves a way for properly incorporating dipole-like (zero net
Burgers vector but non-vanishing) dislocation structures, known as the
statistically stored dislocations (SSDs) into macroscopic models of crystal
plasticity in three dimensions. Moreover, the natural transition between
different equilibrium patterns found here may also shed light on understanding
the emergence of the persistent slip bands (PSBs) in fatigue metals induced by
cyclic loads.
