Dislocations are line defects in crystals which are linked to many of the mechanical and electric properties of crystalline materials. In particular, crystals deform plastically by shifting dislocations through the grid which requires a much lower applied force than if we needed to break up lattice bonds in an entire plane:
Since dislocations are defects of the crystal lattice, they should be thought of as an atomistic phenomenon. In fact, they are not well described by continuum models close to the dislocation line. To compensate for this, continuum models typically use a small parameter ε>0 linked to the grid length scale below which the continuum model is assumed to be invalid.
I am interested in the asymptotic motion of dislocations as ε approaches zero. The underlying assumptions of my research are that dislocations move slowly with respect to relaxation processes in the crystal (quasi-static evolution) and that the evolution is energy driven (gradient flow dynamics of some sort).
The Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Plastic Slip
In this article, we show that dislocations moving by the L2-gradient flow of a phase-field energy functional proposed by Koslowski, Cuitiño and Ortiz  for dislocations lying in a plane penetrated by forest dislocations do not approach motion by the gradient flow of the energy limit as computed by Müller and Garroni  . Physically, this extends the validity of the phase-field model to non-monotone loading and suggests that dislocation motion is linked to plastic rather than elastic deformation of materials. The phase-field describes the amount of plastic slip in the direction of a (fixed) Burgers vector/the extra number of half-planes in the top half of the crystal.
The energy under consideration is a Peierls-Nabarro model on a perforated planar domain (i.e. a Modica-Mortola type energy where the gradient term has been replaced by a fractional Sobolev norm of order s=1/2 and the potential is periodic rather than double well). The perforations model dislocations in a different gliding system which inhibit slip in the plane under consideration. The Gamma-limit of the energies is given essentially additively by a perimeter term stemming from the Modica-Mortola energy and a bulk term stemming from the perforations (in the correct scaling limit).
While the gradient flow of the limiting energy would experience a force from the bulk term that acts to decrease slip, the limit of the phase-field evolutions experiences no such force. On the other hand, there exists an energy barrier in the direction of increasing slip. This asymmetry is indicative of plastic behaviour.
Mathematically speaking, we homogenize a fractional order Allen-Cahn equation on perforated domains by constructing sub- and super-solutions trapping the actual gradient flow evolution.
The motion of curved dislocations in three dimensions: Simplified linearized elasticity
In this project, we show that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends on the full deformation gradient rather than its symmetrized version), the force on a dislocation curve by the gradient of the elastic energy asymptotically approaches the mean curvature of the curve (suitably rescaled). We provide rigorous error bounds in Hölder spaces.
As an application, we show that dislocations moving by the gradient flow of the elastic energy converge to dislocations by the gradient flow of the arclength functional as the cutoff radius goes to zero. For technical reasons, the motion law is given by an H1-type dissipation while convergence of the L^2-gradient flows to mean curvature for the L2-gradient flow is shown to hold under additional assumptions on existence and regularity.
The methods developed here are a blueprint for the more physical setting of real isotropic elasticity and extensions are work in progress.