Line tension approximation

When a metal crystal free of applied stress and containing screw dislocation segments is quenched so that supersaturated vacancies are produced, the screw segments are converted into helices by climb. Show that the converted helices can be at equilibrium with a certain concentration of supersaturated vacancies and find an expression for this critical concentration in terms of appropriate parameters of the system. Use the simple line-tension approximation leading to Eq. 11.12. We note that the helix will grow by climb if the vacancy concentration in the crystal exceeds this critical concentration and will contract if it falls below it. 

Show that regardless of the orientation of a straight dislocation line and its Burgers vector, there will exist a stress system that will convert the dislocation line into a helix whose axis is along the position of the original dislocation when the point-defect concentration is at the equilibrium value characteristic of the stress-free crystal. Use the simple line-tension approximation leading to Eq. 11.12. 

An alternative to the full machinery of elasticity that is especially useful in attempting to make sense of the complex properties of three-dimensional dislocation configurations is the so-called line tension approximation. The line tension idea borrows an analogy from what is known about the perturbations of strings and surfaces when they are disturbed from some reference configuration. For example, we know that if a string is stretched, the energetics of this situation can be described via... 

We will resort to the line tension approximation repeatedly since as was noted above, much may be learned about the key features of a given problem on the basis of such arguments, which are largely geometrical. Again, what is especially appealing about the line tension approach is the prospect for making analytic headway on fully three-dimensional problems. 

Even at the level of the line tension approximation, this equation may be solved to various levels of approximation. In the limit in which the bow-out is small (i.e. u 1), the denominator in the expression given above can be neglected with the... 

In the treatment above, we have used the line tension approximation to deduce both an exact and an approximate treatment of bow-out. The comparison between... 

Fig. 8.31. Comparison of the bow-out geometry resulting from exact (full line) and approximate (dashed line) treatment of the line tension approximation. / is a dimensionless stress such that / = abl/2T.

Similarly, within the line tension approximation, the energy of the state in which a junction of length Ij has formed is given by... 

Energy of Curved Dislocation in Line Tension Approximation We wrote down the energy of a curved dislocation as... 

In considering the energetics of extended defects, we have repeatedly resorted to locality assumptions as well. In particular, in the context of dislocations we have invoked the line tension approximation to assign an energy of configuration to a dislocation of the form... 

Postmortem TEM characterization of the deformation substructures can be performed after any type of mechanical tests. Such characterizations can be used to determine the stresses experienced by dislocations that were frozen in at the end of a test. In the local line tension approximation and for elastically isotropic materials, the dislocation curvature R under a stress x can be derived from [67] ... 

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