Defects, singularities and waves

Abstract

Crystal defects such as dislocations are the basis of macroscopic properties such as the strength of materials and control their mechanical, optical and electronic properties. In recent times, advances in electronic microscopy have allowed imaging of atoms and therefore to visualize the core of dislocations, cracks, and so on. In continuum mechanics, dislocations are treated as source terms proportional to delta functions supported on the dislocation line. Cores and crystal structure are not properly considered and it is hard to describe the motion of crystal defects. Unlike defects in fluids (such as vortices), dislocations move only within glide planes, not in arbitrary directions, and they move only when the applied stress surpasses the Peierls stress, which is not infinitesimal. We have proposed a discrete model describing defects in crystal lattices with cubic symmetry and having the standard linear anisotropic elasticity (Navier equations) as its continuum limit. Moving dislocations are traveling waves which become stationary solutions if the applied stress falls below the Peierls value. The corresponding transition is a global bifurcation of the model equations similar to that observed in simpler one-dimensional Frenkel-Kontorova models. Discrete models can also be used to study the interaction of dislocations and the creation of dislocations under sufficient applied stress.