Dislocations elastic models

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

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. 

Linear Elastic Models of Junction Formation. Yet another scheme for addressing the structure and energetics of dislocation junctions is using the machinery of full three-dimensional elasticity. Calculations of this type are especially provocative since it is not clear a priori to what extent elastic models of junction formation will be hampered by their inability to handle the details of the shortest range part... 

An example of the type of data associated with solution hardening it is the mission of our models to explain was shown in fig. 8.2(a). For our present purposes, there are questions to be posed of both a qualitative and quantitative character. On the qualitative side, we would like to know how the presence of foreign atoms dissolved in the matrix can have the effect of strengthening a material. In particular, how can we reconcile what we know about point defects in solids with the elastic model of dislocation-obstacle interaction presented in section 11.6.2. From a more quantitative perspective, we are particularly interested in the question of to what extent the experimental data permit a scaling description of the hardening effect (i.e. r oc c") and in addition, to what extent statistical superposition of the presumed elastic interactions between dislocations and impurities provides for such scaling laws. 

A second major difficulty with the Peierls model is that it is elastic and therefore conservative (of energy). However, dislocation motion is nonconservative. As dislocations move they dissipate energy. It has been known for centuries that plastic deformation dissipates plastic work, and more recently observations of individual dislocations has shown that they move in a viscous (dissipative) fashion. 

Early in the history of crystal dislocations, the lack of resistance to motion in pure metal-like crystals was provided by the Bragg bubble model, although it was not taken seriously. By adjusting the size of the bubbles in a raft, it was found that the elastic behavior of the raft could be made comparable with that of a selected metal such as copper (Bragg and Lomer, 1949). In such a raft, it was further found that, as expected, the force needed to form a dislocation is large. However, the force needed to move a bubble is too small to measure. 

An elastic continuum model, which takes into account the energy of bending, the dislocation energy, and the surface energy, was used as a first approximation to describe the mechanical properties of multilayer cage structures (94). A first-order phase transition from an evenly curved (quasi-spherical) structure into a... 

Structure no longer relax through an elastic deformation but through the formation of angular gaps or dislocations. For sizes below 100 A, no significant experiments have been performed so far. However, it is expected that the structure should then be that of the relaxed MIC models described in Section IV.B as long as defects or dislocations do not appear. 

Frenkel and Kontorova were not the hrst ones to use the model that is now associated with their names. However, unlike Dehlinger, who suggested the model [99], they succeeded in solving some aspects of the continuum approximation. Like the PT model, the FK model was first used to describe dislocations in crystals. Many of the recent applications are concerned with the motion of an elastic object over (or in) ordered [100] or disordered [101] structures. The FK model and generalizations thereof are also increasingly used to understand the friction between two solid bodies. 

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