Frank Read Source Of Dislocation Pdf

[Klacee Sawatzky]

Inorder to explain the plastic behaviour of a single crystal, a mechanism by which dislocations are generated must be formulated. Such a mechanism is realised on the basis of the two experimental observations:

1. Surface displacement at a slip band is due to the movement of about 1000 dislocations over the slip plane. The number of dislocation sources initially present tin a metal could not account for the observed slip-band spacing and displacement unless there were some way in which each source could produce large amounts of slip before it became immobilized.

The simulation below allows you to explore the effect of changing each parameter in the above equation on the minimum stress. Note that d is changed by changing the (forest) dislocation density (here we model the forest dislocations as pinning sites).

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In a crystalline solid under mechanical stress, a Frank-Read source is a pinned dislocation segment that repeatedly bows and detaches, generating concentric dislocation loops. We demonstrate that, in nematic liquid crystals, an analogous Frank-Read mechanism can generate concentric disclination loops. Using experiment, simulation, and theory, we study a disclination segment pinned between surface defects on one substrate in a nematic cell. Under applied twist of the nematic director, the pinned segment bows and emits a new disclination loop which expands, leaving the original segment intact; loop emission repeats for each additional 180 of applied twist. We present experimental micrographs showing loop expansion and snap-off, numerical simulations of loop emission under both quasistatic and dynamic loading, and theoretical analysis considering both free energy minimization and the balance of competing forces. We find that the critical stress for disclination loop emission scales as the inverse of segment length and changes as a function of strain rate and temperature, in close analogy to the Frank-Read source mechanism in crystals. Lastly, we discuss how Frank-Read sources could be used to modify microstructural evolution in both passive and active nematics.

Nematics are a structured liquid phase of rod-shaped molecules that align with each other, creating orientational order. Such materials can contain defect lines, called disclinations, where that order is disrupted. We have discovered that in a nematic under a twisting deformation, a pinned disclination can bow outward and snap off new loops. These results demonstrate that the Frank-Read mechanism arises in structured fluids, in close analogy to the mechanism in crystalline solids.

This fascinating connection between two very different types of matter represents a fundamental contribution to the field of materials science, and suggests some interesting potential applications. Frank-Read sources form randomly in metals, but in nematics, we can design and build them in specific locations by inscribing defect pinning points on confining walls. This technique controls precisely where defect loops will be generated under shear, a method potentially useful for new liquid-crystal devices.

(a) Geometry for the theoretical free energy of the Frank-Read source. The thick red line represents the curved disclination, viewed from above. (b) Stability diagram in terms of angle δϕ and dimensionless parameter α, indicating whether the curved disclination is stable, metastable, or unstable (so that it must grow to infinity). Along the gray dashed line, the shape is a semicircle. (c) Comparison of theoretical predictions with nematic-order-tensor simulation results, for different defect spacings w.

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In the figure below, two cylindrical holes are carved out to serve as the Frank-Read source. The atoms and elements in figure (b) are colored by the atomic and nodal energy, respectively, and are sliced on the xzxz plane to highlight the holes. In the hybrid simulation, an edge dislocation is first created between the two holes; then subject to a \gamma_zy\gamma_zy simple shear strain, it bows out and form a dislocation loop, leaving behind another edge dislocation segment between the two holes.

Frank-Read source is a type of dislocation multiplication mechanism. Consider a segment whose ends are pinned (corresponding to nodes in a network, precipitates, or sites where the dislocation leaves the glide plane). Under a certain applied stress the segment bows out by glide. As bow-out proceeds, the radius of curvature of the line decreases and the line-tension forces tending to restore the line to a straight configuration increase. For stress less than a critical value, a metastable equilibrium configuration is attained, in which the line-tension force balances that caused by the applied stress. For the large bow-out case, following equilibrium condition holds:

When the net local resolved shear stress( the applied stress plus the internal stresses) exceeds σ * , the loop has no stable equilibrium configuration but passes through the successive positions. Provided that the expanding loop neither jogs out of the original glide plane because of intersections with other dislocations nor is obstructed from rotating about the pinning point, it will annihilate over a portion of its length, creating a complete closed loop and restoring the original configuration. A sequence of loops then continues to form from the source until sufficient internal stresses are generated for the net resolved shear stress at the source to drop below σ * .

3. totalsteps--number of cycles that are run for completion of dd3d command. It should be large enough to secure dislocation configuration has arrived at its equilibrium configuration, for example, 1000 or above.

6. plotfreq--number of cycles between monitered node write statements, higher number makes observation simpler. When the dislocation line goes beyond plotting limits, simply Increasing plim can provide a better view.

8. mobility --mobility law based on which dislocation responds to forces (mobbcc0 for BCC, which is the default setting; mobfcc1 for FCC). For FCC crystals, glide plane is uniquely defined rather BCC, i.e. dislocation motion is confined to a preferred plane.

In the case of Frank-Read source, It is difficult to give a quantitative definition on critical applied stress. When totalsteps is large enough, the bowing process will eventually stop unless that critical stress has been reached. Let's define a certain geometric configuration beyond which the dislocation line becomes unstable and keeps growing, even forming loops. When such configuration is formed, we denote the current external applied stress as the critical one. It is convenient to use maximum distance criteria, that we can extract the coordinates of all nodes from rn and find out the maximum distance from the center of initial dislocation line. If the maximum distance exceeds an empirical value, let's say, L, which is the length of initial dislocation line, we can regard the configuration as "critical" and determine the critical stress, σ * . Several trials have proven that L is a safe one. As long as totalsteps is large enough, 2L, 4L etc will also work well.

The critical stress is determined digit by digit. Assuming the critical externally applied stress is 2.35e-4 times a matrix A, to determine the first digit, simulation is run for 400 steps when the applied stress is 1e-4*A. The stress will be increased by 1e-4*A unless the maximum criteria is not reached. We will see, the first digit is 3. As for the second digit, the initial applied stress starts from 2.1e-4*A and simulation is run for 1000 steps. In the similar manner, 2.2e-4*A will be applied if critical criteria is not met. Finally, the second digit is 4. To determine the third digit, simulation starts at 2.31e-4*A...Of course, the matrix A and totalsteps are subject to change. More simulation steps will be needed when the stress increase is smaller, i.e. better accuracy. For detailed coding, please refer to crstrfr.m.

Based on the maximum distance criteria, we can easily determine the critical applied stress for different initial straight line lengths. Here is a graph showing the relationship between σc and L at a=50.

The relative error decreases from 0.26 to 0.20 when the initial segment coordinate parameter l goes from 1000 to 10,000; however, the error can't be negligible. Better convergence should occur at a smaller a value. Numerical tests shows that, when a=1, relative error decreases from 0.20 to 0.18 when l goes from 1000 to 6000. Lower error is expected for longer segments.

Above results are based on the setting that lmax=l, lmin=l/10 other than the default lmax=1000, limin=200. Though the dislocation line is less smoother, numerical results are the same for the case l=5000. For computation simplicity, we can set lmax and lmin proportional to l and get reasonable results.

We will be using the ParaDiS code for discrete dislocation dynamics in FCC Al. We will explore this using both the compiled C-code version of ParaDiS as well as a simpler MATLAB implementation called DDlab to see dislocation dynamics in action. These codes are freely available (after signup) from the ParaDiS website; there are copies of these on the EWS linux workstations, including the compiled versions of the ParaDiS executables. There is also documentation for both available on EWS in the /class/mse404pla/ParaDiS/doc directory. The directories of interest are:

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