  We derive a phase field crystal model that couples the diffusive evolution of
a microscopic structure with the fast dynamics of a macroscopic velocity field,
explicitly accounting for the relaxation of elastic excitations. This model
captures better than previous formulations the dynamics of complex interfaces
and dislocations in single crystals as well as grain boundary migration in
poly-crystals where the long-range elastic field is properly relaxed. The
proposed model features a diffusivity that depends non-linearly on the local
phase. It induces more localized interfaces between a disordered phase
(liquid-like) and an ordered phase, e.g., stripes or crystal lattices. For
stripes, the interface dynamics are shown to be strongly anisotropic. We also
show that the model is able to evolve the classical PFC at mechanical
equilibrium. However, in contrast to previous approaches, it is not restricted
to a single-crystal configuration or small distortions from a fixed reference
lattice. To showcase the capabilities of this approach, we consider a few
examples, from the annihilation of dislocation loops in a single crystal at
mechanical equilibrium to the relaxation of a microstructure including
crystalline domains with different orientations and grain boundaries. During
the self-annihilation of a mixed type dislocation loop (i.e., not shear or
prismatic), long-range elastic effects cause the loop to move out of plane
before the annihilation event.
