The Line Tension Approximation

In many instances, the evaluation of the complete set of interactions between several dislocations can be prohibitive, at least at the level of the kind of analysis that one might wish to make with a stick in the sand. Indeed, though in the vast majority of this chapter we have been engaged in the development of the elastic theory of dislocations, we now undertake a different framework in which the complex interactions between various infinitesimal segments of a given dislocation are entirely neglected. 

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 

Similarly, if the surface area of a particular interface is altered, the energetics may be handled through a similar idea, namely. 

The analogy with the interfacial energy is more appropriate since dislocations, like free surfaces, have an anisotropy which leads to the conclusion that the energy cost for a small excursion from some flat reference state depends upon the orientation of that reference state. However, the analogy between dislocations and interfaces is imperfect since models of this type have an inherent locality assumption which is exceedingly fragile. To explore this further, we note that in 

Note that we have exploited a parameteric representation of the dislocation line in which the position of the line is given by x(x). 

---

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... 

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... 

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. 

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

Lest the reader think that 2-D foams are just figments of the imagination, it must be pointed out that they can be generated - or at least closely approximated - by squeezing a 3-D foam between two narrowly spaced, wetted, transparent plates (2, 31-35). Structurally even closer realizations may be obtained in phase-coexistence regions of insoluble monolayers of surface-active molecules at the air-water interface (36), where the role of surface tension is taken over by the line tension at the phase boundaries.]... 

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] ... 

In the following text, we use the same approach as in the previous sections of this chapter, which takes into account the interlayer thickness and the effect of the transition zone between the thin interlayer and the bulk liquid. This effect is equivalent to the line tension that is considered in Section 2.10. A low slope and constant surface tension approximations are used. Then, as was shown earlier in Section 2.1 through Section 2.3, it is possible to use the equation taking into account both the disjoining pressure and the capillary pressure in the interlayer. 

Following the principles of the Petrie model, and recalling that the film thickness <5 is much smaller than the radius S/R thin-film approximation, which implies that field equations are averaged over the thickness and that there are no shear stresses and moments in the film. The film is regarded, in fact, as a thin shell in tension, which is supported by the longitudinal force Fz in the bubble and by the pressure difference between the inner and outer surfaces, AP. We further assume steady state, a clearly defined sharp freeze line above which no more deformation takes place and an axisymmetric bubble. Bubble properties can therefore be expressed in terms of a single independent spatial variable, the (upward) axial position from the die exit,2 z. The object... 

---