Dislocation (1D defects)

Dislocation basics

Burgers vector, tangent vector, and slip plane

Burgers vector $\vec b$
- The same dislocation shares the same $\vec b$
Unit tangent vector $\vec t$
Slip plane $\vec b \times \vec t$

Edge dislocation

Edge dislocation $\vec b \perp \vec t$
Edge dislocation symbol $\perp$
- Vertical line: half plane, broken bonds
- Horizontal line: slip plane

Screw dislocation

Screw dislocation $\vec b \parallel \vec t$

Different types of dislocation. via *Science and Design of Engineering Materials* (p.157)

Slip system

Usually, the slip system corresponds to the highest-density plane.

Slip planes for different crystal systems. via *Science and Design of Engineering Materials* (p.162)

Force on dislocation: Peach-Koehler force

The Peach-Koehler formula gives us the force $\vec f_\text{d}$ on a dislocation line under a stress field $[\sigma]$ :

$\frac{\vec f_\text{d}}{\delta l} = (\sigma \cdot \vec b) \times \hat l$

Properties

The shear stress needs to have a component in the $\vec b$ direction. For pure shear with magnitude $\tau$ , $\sigma \, \vec b$ is perpendicular to $\vec b$ :

$\sigma_{ij} \, b_j = \begin{bmatrix} 0 & \tau \\ \tau & 0 \\ \end{bmatrix} \begin{bmatrix} b_x \\ b_y \\ \end{bmatrix} = \begin{bmatrix} \tau\,b_y \\ \tau\,b_x \\ \end{bmatrix}$

Here, the Burgers vector is $\vec b$ and the direction of the dislocation is given by a unit vector $\hat l$ .

This force controls the movement of the dislocation (line). Because of the dot product, the force/movement will always be perpendicular to the dislocation line.

Strain, stress, and self-energy of dislocation

With the Peach-Koehler formula, we are able to obtain the force on a dislocation line under a stress field. If we have the stress field of another dislocation, we can calculate the force “between” the two dislocations.

Strain field, stress field of a dislocation

Here, $G$ denotes the shear modulus of the system and $\nu$ denotes the Poisson ratio.

Results

	Screw dislocation	Edge dislocation
Strain	$\epsilon_{z\theta} = \frac{b}{4\pi}\left(\frac{1}{r}\right)$
Stress	$\sigma_{z\theta} = \frac{Gb}{2\pi}\left(\frac{1}{r}\right)\sigma_{z\theta} = \frac{Gb}{2\pi}\left(\frac{1}{r}\right)\sigma_{z\theta} = \frac{Gb}{2\pi}\left(\frac{1}{r}\right)$	$\sigma_{z\theta} = \frac{Gb}{2\pi(1-\nu)}\left(\frac{1}{r}\right)$
Self energy	$\frac{Gb^2}{4\pi} \ln\frac{R_\text{max}}{R_\text{c}}$	$\frac{Gb^2}{4\pi(1-\nu)} \ln\frac{R_\text{max}}{R_\text{c}}$

Important points:

Stiffer materials have higher dislocation energy, $U_e \propto G$ .
Comparatively, edge dislocation will have higher energy because of the $\frac{1}{1-\nu}$ term.
Strain falls off as $1/r$ .

Elastic energy of a screw dislocation (via Bailey)