Formulating short-range elastic interactions between dislocations in a continuum framework

Permanent deformations of crystalline materials are known to be carried out by a large
number of atomistic line defects, i.e. dislocations. For specimens on micron scales or above, it
is more computationally tractable to investigate macroscopic material properties based on the
evolution of underlying dislocation densities. However, classical models of dislocation
continua struggle to resolve short-range elastic interactions of dislocations, which are believed
responsible for the formation of various heterogeneous dislocation substructures in crystals. In
this talk, we start with discussion on formulating the collective behaviour of a row of
dislocation dipoles, which would be considered equivalent to a dislocation-free state in
classical continuum models. It is shown that the underlying discrete dislocation dynamics can
be asymptotically captured by a set of evolution equations for dislocation densities along with
a set of equilibrium equations for variables characterising the self-sustained dislocation
substructures residing on a shorter length scale, and the strength of the dislocation
substructures is associated with the solvability conditions of their governing equilibrium
equations. Under the same strategy, a (continuum) flow stress formula for multi-slip systems
is also derived, and the formula resolves more details from the underlying dynamics than the
ubiquitously adopted Taylor-type formulae.