Dislocations multiplication

The dislocation parts at A and B (illustration d) have opposite signs and attract each other until the loop closes (illustration e). The dislocation segment, D-D, which was left behind, repeats the same sequence as described above. The force acting on the dislocation, causing it to curve, is balanced by the line tension (discussed in Sect. 3.3.8). The line tension per unit length in that section is equal to the force acting normally on the dislocation segment. In equilibrium with the applied stress, the relation derived as Eq. (3.74) may be rewritten here as  

Although this sometimes occurs through the operation of Frank-Read sources it is not generally observed. What does generally occur is similar, but more complex. The process is called multiple-cross-glide, and was proposed by Koehler (1952). Its importance was hrst demonstrated experimentally by Johnston and Gilman (1959). In addition to its existence, they showed that the process produces copious dislocation dipoles which are responsible for deformation-hardening. 

Koehler attributed the cross-gliding to thermal activation, but it was found experimentally that it increases with dislocation velocity, which is inconsistent with thermal activation, so Gilman (1997) proposed that it is associated with flutter of screw dislocations caused by phonon buffeting. 

The multiple-cross-glide process does not lead to steady-state dislocation multiplication. It does lead to a proportionality between the dislocation density at N, a given time and the rate of increase of dislocation density, dN/dt, that is, to first order kinetics. Thus, the dislocation density grows exponentialy with time  

Note that the dipole structure in a crystal is stabilized by the applied stress. It becomes unstable when the stress is removed. Thus, the in situ structures of 

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Calculations of this type are carried out for fee, bcc, rock salt, and hep crystal structures and applied to precursor decay in single-crystal copper, tungsten, NaCl, and LiF [17]. The calculations show that the initial mobile dislocation densities necessary to obtain the measured rapid precursor decay in all cases are two or three orders of magnitude greater than initially present in the crystals. Herrmann et al. [18] show how dislocation multiplication combined with nonlinear elastic response can give some explanation for this effect. 

The shock-induced micromechanical response of <100>-loaded single crystal copper is investigated [18] for values of (WohL) from 0 to 10. The latter value results in W 10 Wg at y = 0.01. No distinction is made between total and mobile dislocation densities. These calculations show that rapid dislocation multiplication behind the elastic shock front results in a decrease in longitudinal stress, which is communicated to the shock front by nonlinear elastic effects [pc,/po > V, (7.20)]. While this is an important result, later recovery experiments by Vorthman and Duvall [19] show that shock compression does not result in a significant increase in residual dislocation density in LiF. Hence, the micromechanical interpretation of precursor decay provided by Herrmann et al. [18] remains unresolved with existing recovery experiments. 

Flinn et al. [30] describes an experimental impact technique in which <100)-oriented LiF single crystals ( 8 ppm Mg) are loaded in a controlled manner and the multiplication of screw dislocations is measured. The peak shear stress in this relatively soft material is 0.01 GPa. For shear impulses exceeding approximately 40 dyne s/cm, dislocation multiplication is adequately described by the multiple-cross-glide mechanism [(7.24)] with m = l/bL = (2-4) X 10 m, in reasonable agreement with quasi-static measurement [2]. 

To answer questions regarding dislocation multiplication in Mg-doped LiF single crystals, Vorthman and Duvall [19] describe soft-recovery experiments on <100)-oriented crystals shock loaded above the critical shear stress necessary for rapid precursor decay. Postshock analysis of the samples indicate that the dislocation density in recovered samples is not significantly greater than the preshock value. The predicted dislocation density (using precursor-decay analysis) is not observed. It is found, however, that the critical shear stress, above which the precursor amplitude decays rapidly, corresponds to the shear stress required to disturb grown-in dislocations which make up subgrain boundaries. 

Assume the edge dislocation density to be divided into positive and negative populations, N+ and N, moving only on slip planes at 45° (maximum shear stress) to the planar shock front. For a dislocation multiplication (annihilation) rate M, show that conservation of dislocations requires that... 

The dislocation multiplication law N = (l/L)N i is path-independent i.e., N depends only on y and not on the rate at which the deformation occurs. Show that the multiplication law given by... 

J.E. Flinn, G.E. Duvall, G.R. Fowles, and R.F. Tinder, Initiation of Dislocation Multiplication in Lithium Fluoride Monocrystals Under Impact Loading, J. Appl. Phys. 46, 3752-3759 (1975). 

For superlattices with small modulation wavelength of several nanometres, the dislocation multiplication cannot occur, and the dislocation activity is demonstrated by the movement of individual dislocations from B layer into A layer by stress. The critical shear stress to move a dislocation from B layer into A layer (cta/b) can be given by the Lehoczky theory equation [108] as shown in Fig. 13. Figure 13 also gives the normalized oq as function of tglb. It can be seen that there is no strength enhancement as t Ab, which corresponds to very small layer thickness (< 1 nm), and the disappearance of interfaces due to the diffusion between layer A and layer B. The increases rapidly with the increase of... 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

In metals, the incremental stress of deformation-hardening is often reported to be proportional to the square root of the dislocation density. However, In view of the mechanism of dislocation multiplication, and the subsequent deformation hardening, this is highly unlikely, so this author believes that either the data are faulty, or they are being misinterpreted. 

W. G. Johnston and J. J. Gilman, Dislocation Multiplication in Lithium Fluoride... 

Figure 10.11 Dislocation multiplication in ice during plastic deformation. (Courtesy Dr R.Whitworth)...

To solve the vacancy flux equation between dislocations of opposite sign we have to know the dislocation geometry (distance and orientation) in the lattice as the boundary condition. If we consider as a zeroth order approach only the average distance, a, between the dislocations, even this quantity depends on the applied stress and the functioning of dislocation multiplication. Nevertheless, since about l/b2 vacancies are needed for a climb shift of unit length, we may conclude from Eqn. (14.28) and the vacancy flux that the steady-state climb velocity, >d, of a dislocation with edge character is... 

The Kirkendall effect alters the structure of the diffusion zone in crystalline materials. In many cases, the small supersaturation of vacancies on the side losing mass by fast diffusion causes the excess vacancies to precipitate out in the form of small voids, and the region becomes porous [11], Also, the plastic flow maintains a constant cross section in the diffusion zone because of compatibility stresses. These stresses induce dislocation multiplication and the formation of cellular dislocation structures in the diffusion zone. Similar dislocation structures are associated with high-temperature plastic deformation in the absence of diffusion [12-14]. 

Dislocations are present in the natural states of crystalline materials but they drastically increase in number (expressed as the dislocation density, or dislocation length per unit volume) with plastic deformation as existing dislocations spawn new ones. This dislocation multiplication with plastic flow causes an increase in the number of mutual interactions, which hinders their motion. As a consequence, a shear-stress increase must be... 

Figure 10.9. The Frank-Read dislocation loop mechanism of dislocation multiplication. A shear stress causes the portion of a dislocation that is between two pinned segments to bow outward on the slip plane (indicated by arrows). Eventually, the dislocation loop reaches the configuration shown in the middle. When the two curved segments meet, the dislocation loop is freed and a new loop is formed to continue the process.

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. 

Usually, creep deformation of ice single crystals is associated to a steady-state creep regime, with a stress exponent equal to 2 when basal glide is activated . In the torsion experiments performed, the steady-state creep was not reached, but one would expect it to be achieved for larger strain when the immobilisation of the basal dislocations in the pile-ups is balanced by the dislocation multiplication induced by the double cross-slip mechanism. 

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