Dislocation lines motion

Figure 7.2 The formation of a step on the surface of a crystal by the motion of (a) an edge dislocation and (b) a screw dislocation. Note that for an edge, the dislocation Une moves in the direction of the appUed shear stress t for a screw, the dislocation line motion is perpendicnlar to the stress direction. (Adapted from H. W. Hayden, W. G.

For edge dislocations, dislocation line motion and direction of the applied shear stress are parallel for screw dislocations, these directions are perpendicular. 

Dislocation motion produces plastic strain. Figure 9.4 shows how the atoms rearrange as the dislocation moves through the crystal, and that, when one dislocation moves entirely through a crystal, the lower part is displaced under the upper by the distance b (called the Burgers vector). The same process is drawn, without the atoms, and using the symbol 1 for the position of the dislocation line, in Fig. 9.5. The way in... 

A key feature of the motion of dislocation lines is that the motion is rarely concerted. One consequence is that the lines tend not to be straight, or smoothly curved. They contain perturbations ranging from small curvatures to cusps, and kinks. In covalent crystals where there are distinct bonds between the top... 

Dislocation motion in covalent crystals is thermally activated at temperatures above the Einstein (Debye) temperature. The activation energies are well-defined, and the velocities are approximately proportional to the applied stresses (Sumino, 1989). These facts indicate that the rate determining process is localized to atomic dimensions. Dislocation lines do not move concertedly. Instead, sharp kinks form along their lengths, and as these kinks move so do the lines. The kinks are localized at individual chemical bonds that cross the glide plane (Figure 5.8). 

The crystal structure of NiAl is the CsCl, or (B2) structure. This is bcc cubic with Ni, or A1 in the center of the unit cell and Al, or Ni at the eight comers. The lattice parameter is 2.88 A, and this is also the Burgers displacement. The unit cell volume is 23.9 A3 and the heat of formation is AHf = -71.6kJ/mole. When a kink on a dislocation line moves forward one-half burgers displacement, = b/2 = 1.44 A, the compound must dissociate locally, so AHf might be the barrier to motion. To overcome this barrier, the applied stress must do an amount of work equal to the barrier energy. If x is the applied stress, the work it does is approximately xb3 so x = 8.2 GPa. Then, if the conventional ratio of hardness to yield stress is used (i.e., 2x3 = 6) the hardness should be about 50 GPa. But according to Weaver, Stevenson and Bradt (2003) it is 2.2 GPa. Therefore, it is concluded that the hardness of NiAl is not intrinsic. Rather it is determined by an extrinsic factor namely, deformation hardening. 

The stress needed to move a dislocation line in a glassy medium is expected to be the amount needed to overcome the maximum barrier to the motion less a stress concentration factor that depends on the shape of the line. The macro-scopic behavior suggests that this factor is not large, so it will be assumed to be unity. The barrier is quasi-periodic where the quasi-period is the average mesh size, A of the glassy structure. The resistive stress, initially zero, rises with displacement to a maximum and then declines to zero. Since this happens at a dislocation line, the maximum lies at about A/4. The initial rise can be described by means of a shear modulus, G, which starts at its maximum value, G0, and then declines to zero at A/4. A simple function that describes this is, G = G0 cos (4jix/A) where x is the displacement of the dislocation line. The resistive force is then approximately G(x) A2, and the resistive energy, U, is ... 

The disruption to the crystal introduced by a dislocation is characterized by the Burgers vector, b (see Supplementary Material SI for information on directions in crystals). During dislocation motion individual atoms move in a direction parallel to b, and the dislocation itself moves in a direction perpendicular to the dislocation line. As the energy of a dislocation is proportional to b2, dislocations with small Burgers vectors form more readily. 

An edge dislocation is confined to move on its slip plane (conservative motion), and the slip due to the motion of the dislocation is also confined to the slip plane. Movement of a screw dislocation can capture on the plane where it started or else move to any other, parallel to the dislocation line (cross slip). If an edge dislocation were to move... 

Dislocations are line defects. They bound slipped areas in a crystal and their motion produces plastic deformation. They are characterized by two geometrical parameters 1) the elementary slip displacement vector b (Burgers vector) and 2) the unit vector that defines the direction of the dislocation line at some point in the crystal, s. Figures 3-1 and 3-2 show the two limiting cases of a dislocation. If b is perpendicular to s, the dislocation is named an edge dislocation. The screw dislocation has b parallel to v. Often one Finds mixed dislocations. Dislocation lines close upon themselves or they end at inner or outer surfaces of a solid. 

Of particular interest in kinetics is the non-conservative dislocation motion (climb). The net force on a dislocation line in the climb direction (per unit length) consists of two parts Kei is the force due to elastic interactions (Peach-Koehler force), Kcbcm is the force due to the deviation from SE equilibrium in the dislocation-free bulk relative to the established equilibrium at the dislocation line. Sites of repeatable growth (kinks, jogs) allow fast equilibration at the dislocation. For example, if cv is the supersaturated concentration and c is the equilibrium concentration of vacancies, (in the sense of an osmotic pressure) is... 

Fig. 3. Effect of motion of a dislocation line in calcite on dissolution. The crystal was etched in cone, formic acid (15 sec) then stressed at 500°C for 5 min, and finally etched again (15 sec). During the stressing one of the dislocations moved. X500, before reduction for publication.

Kinematics of Lines. The first order of business is the characterization of the dislocation lines themselves without reference to the forces that result in dislocation motion. In principle, each and every dislocation line can be characterized through a parameterization of the form... 

Fig. 17) onto an intersecting slip plane (see Fig. 23). After travelling a short distance on the new plane, that part of the dislocation may resume its motion on another plane of the original slip system. The segment XY, which has reached a third slip plane, can now act as a Frank-Read source and radiate new loops on their respective planes. In other words, through a process of cross glide, a dislocation line can make its own Frank-Read source.

If one considers dislocations in more detail, it becomes clear that not just the atoms in the dislocation line are displaced but also the neighboring atoms. For example, consider the edge dislocation shown in Fig. 6.7. The plane of atoms below and above the slip plane are shown. In the initial configuration, several atoms in the upper row (five) are displaced from their normal positions, i.e., there is a dislocation width. It is useful to compare this figure with Fig. 6.2 in which the displacement b is obtained by simultaneous motion of all the atoms. The same displacement b is obtained in both cases but for the dislocation it is produced by a localized motion of atoms rather than the simultaneous shear of a perfect plane. Thus, the displacement associated with the dislocation is spread across... 

Plastic deformation results from the accumulated motion of numerous dislocations at the atomic scale. The dislocation density p is a parameter representing the average amount of accumulated plastic deformation or, in other words, the amount of deviation from the strictly geometrical lattice structure. The dislocation density p is defined as the total length of dislocation lines per cubic centimeter, and it is almost identical to the flow stress of a metal under hot forming or the internal stress Oi. Numerous dislocations are introduced by forming. For example, the initial dislocation density of a fully annealed structure po is about 10 (cm/cm ) but increases to lO (cm/cm ) or more after metal forming. 

The motion of an edge dislocation is restricted to one plane only-the glide (or slip) plane. Both a positive and a negative edge dislocation can only exit a glide plane by means of the climb process, as illustrated in Fig. 3.39. For climb to happen in a positive dislocation (the upper illustrations in Fig. 3.39), vacancies, arriving by diffusion, must occur adjacent to the dislocation line by replacing one... 

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