Dislocation Formation and Work-Hardening in Two-Phase Alloys
  A phase field model is presented to investigate dislocation formation (coherency loss) and workhardening in two-phase binary alloys. In our model the elastic energy density is a periodic function of the shear and tetragonal strains, which allows multiple formation of slips (dislocation dipoles). By numerically integrating the dynamic equations in two dimensions, we find that dislocations appear in pairs in the interface region and grow into slips. One end of each slip glides preferentially into the softer region, while the other end remains trapped at the interface. Under uniaxial stretching at deep quenching, slips appear in the softer regions and do not penetrate into the harder domains, giving rise to an increase of the stress with increasing applied strain in plastic flow. 
Akihiko Minami and Akira Onuki