Dislocation Bow-Out

Our interest is in determining the energy minimizing configuration of the bowed-out segment. To do so, we note that this has become a simple problem in variational calculus, with the relevant Euler-Lagrange equation being 

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 

The solution to this equation can be written down by inspection as 

The equation for bow-out may also be solved exactly without resorting to the approximation in which m (x) 1. In this case, the equation of equilibrium may be written (after performing the first integration) 

Like its approximate partner derived above, this equation can be rewritten in dimensionless variables as 

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Fig. 8.30. Geometry associated with dislocation bow-out between two pinning points. 

Recall from the discussion on line tension and dislocation bow-out given in... 

Fig. 14. (a) Weak beam dark-field image of dislocations in biotite. The trace of the slip plane (S) is indicated as well as dislocations bowing out (X). [346]. (b) Dark-field image of screw dislocations in... 

Fig. 11. Operation of a Frank-Read Source. In (a) the dislocation leaves the slip plane at B and C, which act as anchoring points. Under an applied stress the dislocation segment BC expands, finally forming a loop around BC, leaving a dislocation segment between B and C. (e) The dislocation bowed out just before the loop breaks loose from the pinning points from Weertman and Weertman and Read. ...

Figure 7.12. Bowed-out dislocation segment under influence of applied shear stress bG/R in the shock-compressed state.

Figure 3.13 Dislocation pinned at each end (a) can respond to stress by bowing out (b, c, d, and e) to form a dislocation loop and reform the pinned dislocation (/). The growth of the dislocation represented in (a)-(c) requires increasing local stress, whereas the steps in (d)-(f) are spontaneous once the point in (c) is passed.

Figure 11.9 Dislocation segment pinned at A and B bowing out in the slip plane due...

Solution. The source will be able to become active if the driving osmotic climb force is large enough to overcome the restraining curvature force that reaches a maximum when the dislocation segment has bowed out to the minimum radius of curvature corresponding to R = L/2. Setting /M = fK, we then have the critical condition... 

For the special case of the infinitesimal bow-out of a dislocation segment under the achon of the stress ct., we note that... 

The construction of dislocation dynamics methods is predicated upon the discretization of dislocation lines into a series of segments. The goal of this problem is to gain facility in such discretizations. Consider two pinning points at (0, 0) and (L, 0) in the slip plane characterized by z = 0 and assume that the dislocation line between these two points has bowed out and assumed a parabolic bow-out profile. Discretize the bowed segment using N nodes and... 

Consider the bow-out of a pinned segment like that considered in problem 7 in the presence of a shear stress r. Obtain the dynamical equations for the nodes on this line using the line tension dislocation dynamics ideas introduced in this chapter. 

Figure 3.23 A dislocation pinned at each end (part a) can respond to stress by bowing out (parts Iv l) to form a dislocation loop and reform the pirmed dislocation (part e)...

Dislocation multiplication occurs when the dislocations are made to move during deformation. One possible multiplication mechanism is the Frank-Read source. Suppose that a dislocation is pinned at two points, which are a distance / apart, as shown in Figure 12.21a. Under the action of an applied stress the dislocation will bow out. The radius of curvature R is related to the applied shear stress. To. 

At this point, it is constmctive to observe the dislocation structure in MgO polycrystalline ceramics, as revealed by TEM. In Fig. 6.36, a typical dislocation structure is shown for creep deformation, not unlike that found in metals, with the presence of subgrains, in which a 3D dislocation network may be seen. Note that the long dislocation segments seldom run in a straight line from one node to another, but are bowed out, often in only one plane, though, sometimes, as seen in Fig. 6.36a,... 

Fig. 6.36 Typical dislocation structures. In a loops, L, and bowed-out dislocations, b are visible [28], With kind permission of John Wiley and Sons...

Let there be several obstacles in our material with a distance of 2A between them (figure 6.23). Consider a dislocation pinned on these obstacles. When the external stress r acts on the dislocation, it tries to move on and bows out. Its shape is a segment of a circle because this covers the greatest area with the least-most energy to create new length of dislocation line. 

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