7.2: Dislocation Generation
- Page ID
- 8207
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Frank-Read source
In order 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.
2. If there were no source generating dislocations, cold-work should decrease, rather than increase, the density of dislocations in a single crystal.
The mechanism by which dislocations are generated (multiplied) was proposed by Frank and Read in the 1950s and is known as the Frank-Read source.
Below is a video which explains how dislocations are generated from a Frank-Read source.
Animation captions:
- When a tensile stress is applied to a single crystal, the shear stress exerts a force \(F = \tau * b\) on the dislocation line, which is pinned at both ends. This could occur if the two ends were nodes where the dislocation in the plane of the paper intersects dislocations in the other slip planes, or the pinning could be caused by existing precipitates
- The shear stress causes the dislocation line to bow outwards, balancing the line tension and the force due to the applied shear stress.
- The shear stress reaches a maximum when the segment becomes a semicircle.
- The dislocation segment continues to expand until the ends annihilate each other (as they have opposite burgers vectors), forming a dislocation loop.
- The loop continues to expand, and a new dislocation line is formed. The process can repeat itself sending out many loops.
Here is a TEM video showing a real Frank-Read source in action:
The minimum stress required to operate a Frank-Read source
Figure 2: Force diagram of a Frank-Read source. The dislocation segment is pinned at both ends by forest dislocations.
The force on the dislocation line, which has a distance d between the pinned ends, when a shear stress τ is applied is \(F= \tau b d\). This force is balanced by the line tension (energy/length) of the dislocation, which is ≈ Gb2.
At the pinning ends, the vertical component of the force is 2Gb2sin(θ). This force reaches a maximum 2Gb2 when the dislocation is bowed into a semicircle (θ) = 90). Hence, the minimum stress required to operate the Frank-Read source is when τbd = 2Gb2, i.e.
\[\tau = 2 \frac{Gb}{d} \]
Since the distance d between the pinned ends is related to the dislocation density ρ by \(\rho = 1 / d^2 \), the minimum stress can be written:
\[\tau = 2Gb \sqrt{\rho}\]
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).