Peierls—Nabarro model

The crucial argument in the PN theory is that the elastic energy of the dislocation core is balanced by the energy cost of introducing the misfit in the lattice. In the following discussion we will drop, for the reasons we mentioned earlier, the infinite term which appears in all the expressions we derived above. We will 

A variational derivative of this expression with respect to the dislocation density 

The treatment presented here does not follow the traditional approach of guessing a sinusoidal expression for the shear stress (see for example the treatment by Hirth and Lothe, mentioned in the Further reading section). Instead, we adopt a more general point of view based on a variational argument for the total energy of the dislocation, and introduce the sinusoidal behavior as a possible simple choice for the displacement potential. The essence of the resulting equations is the same as in traditional approaches. 

The first term is the elastic stress at point x due to the infinitesimal dislocation p(x )dx at point x the second term represents the restoring stress due to the non-linear misfit potential acting across the slip plane. This potential must be a periodic function of u(x) with a period equal to the Burgers vector of the dislocation. As the simplest possible model, we can assume a sinusoidal function for the misfit potential, which is referred to as the Frenkel model [136]. One choice is (see Problem 4 for a different possible choice) 

A typical dislocation profile is shown in Fig. 10.9. In this figure, we also present the dislocation profiles that each term in Eq. (10.20) would tend to produce, if acting alone. The elastic stress term would produce a very narrow dislocation to minimize the elastic energy, while the restoring stress would produce a very broad dislocation to minimize the misfit energy. The resulting dislocation profile is a compromise between these two tendencies. 

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A more elegant (and mathematically tractable) description of the problem of dislocation pile-ups is to exploit the representation of a group of dislocations as a continuous distribution. This type of thinking, which we have already seen in the context of the Peierls-Nabarro model (see section 8.6.2), will see action in our consideration of cracks as well. The critical idea is that the discrete set of dislocations is replaced by a dislocation density pb (x) such that... 

Whereas the line tension was invoked as a way to capture the self-energy of dislocations from an elastic perspective, there are also ways of capturing core effects on the basis of locality assumptions. Recall that in our treatment of dislocation cores we introduced the Peierls-Nabarro model (see section 8.6.2) in which the misfit energy associated with slip displacements across the slip plane is associated with an energy penalty of the form... 

Lu G, The Peierls-Nabarro model of dislocations A venerable theory and its current development. Yip S, editor, Handbook of Materials Modeling. Volume I Methods and Models, Netherlands ... 

Beltz, G. E. and Freund, L. B. (1994), Analysis of the strained layer critical thickness concept based on a Peierls-Nabarro model of a threading dislocation. Philosophical Magazine 69, 183-202. 

Figure 10.9. Profile of an edge dislocation Top the disregistry or misfit u(x) as dictated by the minimization of the elastic energy (solid line) or the misfit energy (dashed line) and the corresponding densities p(x), given by Eq. (10.10). Bottom the disregistry and density as obtained from the Peierls-Nabarro model, which represents a compromise between the...

The Peierls-Nabarro model has been used to determine properties of dislocation cores, the misfit energy and particularly changes with pressure. This is based on the assumption of a planar core which is the most able to ghde. It has direct implications for slip systems. In order to move, a dislocation must overcome an energy barrier under an applied stress. The Peierls-Nabarro model has been used to constrain dislocation core sizes and Peierls stresses in several oxides and sihcates relevant to the Earth s mantle, particularly periclase [439], ohvine [440,441], ringwoodite [80],... 

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