Dislocations edit page

Dislocation are microscopic displacements within the regular atom lattice of a crystalline material usually as a result of plastic deformation. Dislocations are described by a Burgers vector describing the direction of the atomic shift and a line vector describing the direction of the displacements within the material. One distinguishes two cases:

## Edge Dislocations

Here the directions of the atomic shifts are orthogonal to the direction the displacements spread within the material. In order to define a edge dislocation we proceed as follows

## Screw Dislocations

Screw dislocations are characterized by the fact that Burgers vector and line vector are perpendicular to each other.

## Relation to Slip Systems

Dislocation systems are tightly related to slip systems. Given a set of slip systems the corresponding edge and screw dislocations can be computed by

A shortcut for the above lines is

## The Dislocation Tensor

As each dislocation corresponds to an deformation of the atom lattice a dislocation can also be described by a deformation matrix. This matrix is the dyadic product between the Burgers vector and the line vector and can be computed by

Note that the unit of this tensors is the same as the unit used for describing the length of the unit cell, which is in most cases Angstrom (au). For amount of deformation the norm of the Burgers vectors is important

## The Energy of Dislocations

The energy of each dislocation system can be stored in the property u. By default this value it set to 1 but should be changed according to the specific model and the specific material.

According to Hull & Bacon the energy U of edge and screw dislocations is given by the formulae

$U_{\mathrm{screw}} = \frac{Gb^2}{4\pi} \ln \frac{R}{r_0}$

$U_{\mathrm{edge}} = (1-\nu) U_{\mathrm{screw}}$

where

• G is
• b is the length of the Burgers vector
• nu is the Poisson ratio
• R
• r

In this example we assume

R = r_0 = U = norm(dS.b).^2