Dislocation dynamics

If this segment has an edge component of the right sign, it may provide strain relief When the critical thickness is exceeded, the force resulting from strain relief exceeds the force necessary for dislocation extension (the Peach Koehler force [9]) and a misfit relief segment is created. 

The dislocation loop nucleation energy increases as the dislocation energy increases. Thus, relatively soft materials such as II-VI semiconductors would be more likely to nucleate misfit dislocations by loop formation than would harder materials such as Si. Materials with small misfit strains are less likely to form dislocations by loop expansion, because the loop would have to be much larger before the energy payback from strain relief would compensate for the cost of the loop. Stress concentrators can reduce the loop nucleation barrier, although in many materials such concentrators are rare. If the effective additional stress associated with the concentration point adds to the misfit stress, the local energy barrier to loop nucleation can be reduced or eliminated. 

Examples of stress concentrators include edges of the strained layer on patterned substrates, surface or growth defects (second phases, and other imperfections), or local misorientations of the substrate-film interface relative to a low-index plane. 

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We also want to point out the difference between simple rate-dependent phenomena and path-dependent effects. Simple rate dependence means that the internal micromechanical state (as possibly represented by some meso-scale variables) depends only on the current deformation and current rate of deformation the material has no memory of the past. In terms of dislocation dynamics and (7.1), a simple rate-dependent constitutive description would be one in which... 

J.W. Taylor, Dislocation Dynamics and Dynamic Yielding, J. Appl. Phys. 36, 3146-3150(1965). 

Dremin, A.N. and Breusov, O.N., Processes Occurring in Solids Under the Action of Powerful Shock Waves, Russian Chem. Rev. 37 (5), 392-402 (1968). Gilman, J.J., Dislocation Dynamics and the Response of Materials to Impact, Appl. Meek Rev. 21 (8), 767-783 (1968). 

From the early work of Taylor [63T01] connecting dislocation behavior to observed viscoplastic shock-compression response, numerous studies have attempted to relate conventional dislocation dynamics models to experimental observations. Theory and observations consistently require unusually large numbers of mobile dislocations. Although qualitatively descriptive, progress to date on dislocation models has not proven to provide quantitative descriptions to the observations in metals. 

M.J. Luton, J.J. Jonas, "A model for high temperature deformation based on dislocation dynamics, rate theory and a periodic intermnal stress" Acta Met. 18, 511, 1970... 

A. V. Granato, Internal Friction Studies of Dislocation Motion, p. 117, in Dislocation Dynamics, Edited by A. R. Rosenfield, G. T. Hahn, A. L. Bement, and R. I. Jaffee, McGraw-Hill Book Company, New York (1968). 

The non-monotonous dependence of surface layer microhardness on deformation degree results from different mechanisms of nitrogen diffusion in deformed material. In our point of view, under the deformations of 3-8 and 20-30 % the greatest number of mobile dislocations, capable to provide the additional transfer of nitrogen interstitial atoms with Cottrell s atmospheres by the dislocation-dynamic mechanism [6-8], can be formed. 

Haasen, P. In Dislocation Dynamics Rosenfield, A. R. Hahn, G. T. Bement, A. L. Jaffee, I., Eds. Battelle Institute Materials Colloquia Columbus, 1967. 

Preliminary Dislocation Dynamics (DD) simulations using the model developed by Verdier et al. provide a plausible scenario for the dislocation patterning occuring during the deformation of ice single crystals based on cross-slip mechanism. The simulated dislocation multiplication mechanism is consistent with the scale invariant pattemings observed experimentally. 

Fig. 8.41. Frank-Read source as obtained using dislocation dynamics (adapted from Zbib et al. (1998)).

Fig. 8.47. Lomer-Cottrell junction as computed using three-dimensional elasticity representation of dislocation dynamics (adapted from Shenoy et al. (2000)).

The final feature of the dislocation dynamics method that must be introduced so as to give such methods the possibility of examining real boundary value problems in plastic deformation is the treatment of boundary conditions. In particular, if we wish to consider the application of displacement and traction boundary conditions on finite bodies, the fields of the relevant dislocations are no longer the simple infinite body Volterra fields that have been the workhorse of our discussions throughout this book. To confront the situation presented by finite bodies, a useful scheme described in Lubarda et al. (1993) as well as van der Giessen and Needleman (1995) is to use the finite element method to solve for the amendments to the Volterra fields that need to be considered in a finite body. Denote the Volterra fields for an infinite body as In this case the fields of interest are given by... 

Fig. 12.25. Illustration of treatment of boundary conditions for dislocation dynamics analysis of body of finite extent (adapted from Lubarda et al. (1993)). The symbols and " refer to the fields associated with an infinite body and the corrections, respectively.

Line Tension Dislocation Dynamics. Though our discussion above has given a generic feel for the dislocation dynamics strategy and has emphasized the full treatment of interactions between different segments, a watered down version of... 

Fig. 12.26. Results of mesoscopic dislocation dynamics simulation of dislocation interactions in a strained epitaxial layer (adapted from Schwarz and LeGoues (1997)). The network shown in the figure results from the interaction of dislocations on parallel glide planes.

As noted above, one interesting application of these ideas is to the motion of a dislocation through an array of obstacles. An alternative treatment of the field due to the disorder is to construct a particular realization of the random field by writing random forces at a series of nodes and using the finite element method to interpolate between these nodes. An example of this strategy is illustrated in fig. 12.27. With this random field in place we can then proceed to exploit the type of line tension dislocation dynamics described above in order to examine the response of a dislocation in this random field in the presence of an increasing stress. A series of snapshots in the presence of such a loading history is assembled in fig. 12.28. 

Fig. 12.28. Snapshots from the motion of a dislocation through a random stress field as obtained using a line tension variant of the dislocation dynamics method (courtesy of Vivek Shenoy).

An attractive higher-level framework within which to study plasticity during nanoindentation is provided by the dislocation dynamics methods described above. In particular, what makes such calculations especially attractive is the possibility of making a direct comparison between quantities observed experimentally and those computed on the basis of the nucleation and motion of dislocations. In particular, one can hope to evaluate the load-displacement curve as well as the size and shape of the plastic zone beneath the indenter, and possibly the distribution of dislocations of different character. While the... 

Computational Mechanics at the Mesoscale by A. Needleman, Acta Mater. 48, 105 (2000) offers a deeper perspective on the multiscale aspects of modeling plasticity than that offered in the current chapter. Needleman s discussion descends from the perspective of crystal plasticity to the level of dislocation dynamics. 

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