The Peierls-Nabarro Model

This result may be borrowed for the continuous distribution imagined here once we recognize that the Burgers vector of each such infinitesimal dislocation is related to the local gradient in the slip distribution as is seen in fig. 8.27. Hence, the elastic energy due to all such dislocations is given by 

In the equation given above, / is an inconsequential constant introduced as a large distance cutoff for the computation of the logarithmic interaction energy. 

For the special case in which we make the simplifying assumption that the interplanar potential takes the simple cosine form given in eqn (8.2) and advocated earlier in the context of the ideal strength model, this problem allows for direct analytic solution. In particular, the solution is 

One of the key features that emerges from this solution, and a crucial hint for the type of analyses we will aim for in the future, is the fact that the stresses implied by the Peierls-Nabarro solution do not suffer from the same singularities that plague the linear elastic solution. A detailed examination of this point may be found in Hirth and Lothe (1992) and is illustrated in their eqn (8-13). By introducing an element of constitutive realism in the form of a nonlinear (and in fact nonconvex) interplanar potential, the solution is seen to be well behaved. 

In particular, to compute the Peierls stress, we make the transcription 

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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 ... 

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],... 

Our treatment in this section will cover three primary thrusts in modeling dislocation core phenomena. Our first calculations will consider the simplest elastic models of dislocation dissociation. This will be followed by our first foray into mixed atomistic/continuum models in the form of the Peierls-Nabarro cohesive zone model. This hybrid model divides the dislocation into two parts, one of which is treated using linear elasticity and the other of which is considered in light of a continuum model of the atomic-level forces acting across the slip plane of the dislocation. Our analysis will finish with an assessment of the gains made in direct atomistic simulation of dislocation cores. 

Thus far, our discussion of dislocation core effects has been built around those tools admitting of direct analytic progress such as is offered by the Peierls-Nabarro dislocation core model. This model has many virtues. However, it is also strictly limited, with one of its primary limitations being that without huge machinations, it may only be applied to dislocations with planar cores. To explore the full atomic-level complexity that arises in dislocation cores, we must resort to numerical techniques, and direct atomistic simulation in particular. The remainder of our discussion on dislocation cores will take up the question of how one goes about carrying out such simulations and what may be learned from them. 

Xu, G. and Zhang, C. (2003) Analysis of dislocation nucleation from a crystal surface based on the Peierls-Nabarro dislocation model, J. Mech. Phys. Solids, 51, 1371-1394. 

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. 

Fig. 8.26. Schematic of key elements in Peierls-Nabarro dislocation model (courtesy of R. MiUer). The key idea is the use of a nonlinear constitutive model on the relevant slip plane which is intended to mimic the atomic-level forces resulting from slip.

The P-N model was first proposed by Peierls [22] and Nabarro [23] to incorporate the details of a discrete dislocation core into a framework that is essentially a continuum. More specifically, the atomistic scale information of the P-N model is contained in the form of the... 

At roughly the same time, Peierls (1940), and Nabarro (1947) developed a two-dimensional model of a dislocation in a simple square crystal structure. This model indicated that a small, but finite, amount of energy is needed to... 

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