  An exactly solvable phase-field theory of dislocation dynamics, strain
hardening and hysteresis in ductile single crystals is developed. The theory
accounts for: an arbitrary number and arrangement of dislocation lines over a
slip plane; the long-range elastic interactions between dislocation lines; the
core structure of the dislocations resulting from a piecewise quadratic Peierls
potential; the interaction between the dislocations and an applied resolved
shear stress field; and the irreversible interactions with short-range
obstacles and lattice friction, resulting in hardening, path dependency and
hysteresis. A chief advantage of the present theory is that it is analytically
tractable, in the sense that the complexity of the calculations may be reduced,
with the aid of closed form analytical solutions, to the determination of the
value of the phase field at point-obstacle sites. In particular, no numerical
grid is required in calculations. The phase-field representation enables
complex geometrical and topological transitions in the dislocation ensemble,
including dislocation loop nucleation, bow-out, pinching, and the formation of
Orowan loops. The theory also permits the consideration of obstacles of varying
strengths and dislocation line-energy anisotropy. The theory predicts a range
of behaviors which are in qualitative agreement with observation, including:
hardening and dislocation multiplication in single slip under monotonic
loading; the Bauschinger effect under reverse loading; the fading memory
effect, whereby reverse yielding gradually eliminates the influence of previous
loading; the evolution of the dislocation density under cycling loading,
leading to characteristic `butterfly' curves; and others.
