  The phase-field crystal model in its amplitude equation approximation is
shown to provide an accurate description of the deformation field in defected
crystalline structures, as well as of dislocation motion. We analyze in detail
the elastic distortion and stress regularization at a dislocation core and show
how the Burgers vector density can be directly computed from the topological
singularities of the phase-field amplitudes. Distortions arising from these
amplitudes are then supplemented with non-singular displacements to enforce
mechanical equilibrium. This allows for the consistent separation of plastic
and elastic time scales in this framework. A finite element method is
introduced to solve the combined amplitude and elasticity equations, which is
applied to a few prototypical configurations in two spatial dimensions for a
crystal of triangular lattice symmetry: i) the stress field induced by an edge
dislocation with an analysis of how the amplitude equation regularizes stresses
near the dislocation core, ii) the motion of a dislocation dipole as a result
of its internal interaction, and iii) the shrinkage of a rotated grain. We also
compare our results with those given by other extensions of classical
elasticity theory, such as strain-gradient elasticity and methods based on the
smoothing of Burgers vector densities near defect cores.
