The phase-field crystal model in an 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 stress regularization at a dislocation core given by the model, 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 a 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 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.
Bibliographical noteFunding Information:
We acknowledge M. Lazar and B. Svendsen for fruitful discussions. A.V. acknowledges support from the German Research Foundation under Grant No. Vo899/20 within SPP 1959. K.R.E. acknowledges financial support from the National Science Foundation (NSF) under Grant No. DMR-1506634 , Z.-F.H. acknowledges support from NSF under Grant No. DMR-1609625 , and J.V. acknowledges support from NSF under Grant No. DMR-1838977 . L.A. acknowledges support from Research Council of Norway through CoE funding scheme, Project No. 262644 . We also gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Jülich Supercomputing Centre (JSC), within the Project No. HDR06, and by the Information Services and High Performance Computing (ZIH) at TU Dresden.
- crystal plasticity
- dislocation motion
- finite element method
- phase-field crystal