Chapter 1 introduced dislocations as dual objects permitting both atomistic and continuum descriptions. The subsequent Chapters 2 through 7 discussed various aspects of atomistic simulations and their application to dislocation modeling. In the rest of the book, from Chapter 8 to Chapter 11, we will be treating dislocations as continuum objects. This is a huge simplification that makes it possible to consider dislocation behavior on length and time scales well beyond reach of the atomistic simulations. The following chapters are organized in the order of increasing length and time scales. This particular chapter deals with the famous Peierls–Nabarro continuum model that is most closely related to the atomistic models discussed earlier. Fundamentally, dislocations are line defects producing distortions in an otherwise perfect crystal lattice. While this point of view is entirely correct, the atomistic models of dislocations can deal with only relatively small material volumes where every atom is individually resolved. Furthermore, having to keep track of all these atoms all the time limits the time horizon of atomistic simulations. On the other hand, when the host crystal is viewed as an elastic continuum, the linear elasticity theory of dislocations offers a variety of useful analytical and numerical solutions that are no longer subject to such constraints. Although quite accurate far away from the dislocation center, where the lattice distortions remain small, continuum theory breaks down near the dislocation center, where lattice discreteness and non-linearity of interatomic interactions become important. To obtain a more efficient description of crystal dislocations, some sort of bridging between the atomistic and continuum models is necessary. For example, it would be very useful to have a hybrid continuum–atomistic approach such that it retains the analytic nature of the continuum theory for the long-range elastic fields but also captures the essential non-linear effects in the atomic core. Bearing the names of Rudolf Peierls [86] and Frank Nabarro [87], the celebrated Peierls–Nabarro (PN) model is one such approach. Possibly the most attractive feature of the PN model is its simplicity.
Atomistic simulations of the graded residual elastic fields in metallic nanowires
