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\chapter{Basic Dislocation Theory}
\label{chap:disloc}


This appendix gives a short introduction on dislocation theory. It is aimed to aid the
understanding of the relationship between dislocation motion and AE, as presented in
Section~\ref{sec:ae_generation} and also discussed in Section~\ref{sec:disloc_AE}. It
gives some relevant information for the understanding of some of the equations that
govern dislocation motion during plastic deformation. For more information on the
subject, the literature from which this appendix is based must be
consulted~\cite{hull84,honeycombe84}.

\section{Dislocation Definition}

In a crystalline material, a dislocation represents a line defect or discontinuity
between part of the crystal that has sheared and part which has not. A dislocation line
extends over a macroscopic distance within the material and can form closed loops,
branch into other dislocations, or end at the surface or grain boundary of the crystal.
A dislocation is defined by two parameters (Figure~\ref{fig:disloc_definition}): the
Burgers vector ($b$) and the dislocation line direction. A special plane, called the
slip plane, is defined by these two parameters. Two simple types of dislocations can be
defined:
\begin{figure}[h]
   \center{\includegraphics{disloc_definition.eps}}
   \caption[Dislocation definition.]
   {Dislocation definition. (a)~Pure edge type. (b)~Pure screw
   type. (c)~Mixed type.}
   \label{fig:disloc_definition}
\end{figure}
\begin{description}

   \item [Edge dislocations] The Burgers vector is normal to the dislocation line.

   \item [Screw dislocations] The Burgers vector is parallel to the dislocation line.

\end{description}
However, as shown in Figure~\ref{fig:disloc_definition}c, dislocations normally have a
mixed edge and screw character, where the Burgers vector lies at an arbitrary angle to
the dislocation line direction. A mixed dislocation can be resolved into its edge and
screw components: $b = \sqrt{b^2_\mr{edge} + b^2_\mr{screw}}$.

Two dislocations with opposite Burgers vectors are physical opposites of each other and,
if brought together, they annihilate and restore the perfect crystal.


\section{Dislocation Movement}

There are two basic types of dislocation movement:

\begin{description}

   \item [Glide] In this type of motion, also known as conservative, dislocations
   move on their slip plane. However, since the Burgers vector and line of a screw
   dislocation do not define an unique plane, the glide of this type of dislocation is not
   restricted to a specific slip plane. Glide of many dislocation results in slip, which
   is the the most common manifestation of plastic deformation in crystals.

   \item [Climb] This type of motion is also known as non-conservative, and the
   dislocations move out of the slip plane. Climb requires thermal activation
   since it involves the diffusion of atoms either towards or away from the dislocation,
   and, therefore, it occurs at higher temperatures.

\end{description}

  At a high enough shear stress
acting on the slip plane, a dislocation glides on this plane, and if it reaches the
surface of the crystal, it contributes to the total plastic deformation by $b$. More
generally, if each dislocation on the crystal moves (by glide or climb) an average
distance $\bar x_\mr{D}$, a relationship for the macroscopic plastic shear strain
($\gamma$) can be established:
\begin{eqnarray}
   \gamma  = b\rho_\mr{MD}\bar x_\mr{D}
   \label{eq:disloc_strain}
\end{eqnarray}
where $\rho_\mr{MD}$ is the density of mobile dislocations. And the shear strain rate
($\dot\gamma$) can be calculated from the following equation:
\begin{eqnarray}
   \dot\gamma  = b\rho_\mr{MD}\bar V_\mr{D}
   \label{eq:disloc_strain_rate}
\end{eqnarray}
where $\bar V_\mr{D}$ is the average dislocation velocity. Moreover, an empirical
relationship between $V_\mr{D}$ and the applied stress ($\tau$) has been established:
\begin{eqnarray}
   V_\mr{D} \propto \left( \frac{\tau}{\tau_\mr{0}} \right)^n
   \label{eq:disloc_speed}
\end{eqnarray}

\section{Multiplication of Dislocations}

All crystals contain dislocations, and in well-annealed crystals, dislocations are
arranged in a ill-defined network (Frank net). The density of dislocations ($\rho_\mr{D}$)
in a crystal can be defined in two ways: as the total length of dislocation line per unit
volume, or as the number of dislocations intersecting a unit area, both measured in units
of mm$^{-2}$. In well-annealed metals, $\rho_\mr{D}$ is usually between $10^4$ and
$10^6$~mm$^{-2}$. However, $\rho_\mr{D}$ increases rapidly with plastic deformation. After
large amounts of plastic deformation, $\rho_\mr{D}$ is in the range
$10^8$--$10^9$~mm$^{-2}$, indicating that there must be a process by which dislocations
are generated due to plastic deformation.

A mechanism, referred to as Frank-Read source, involves that a dislocation line is
locked at either one or two points. In both cases, the dislocation line will tend to
move when the shear stress applied to the slip plane ($\tau$) reaches a critical stress
$\tau_0$ that initiates dislocation displacement, but since it is locked at one or two
points, it will rotate around these points. If it is locked at just one point, the
dislocation line will fully rotate around the locking point, generating a displacement
of magnitude and direction of $b$ when the dislocation line eventually reaches the
surface of the crystal, and it will keep rotating until the dislocation line reaches
its initial position. At this stage, the process is free to recommence, and if it takes
place $n$ times, it will produce a displacement $nb$ at the surface of the crystal. If
the dislocation line is locked at two points, the dislocation line will expanded until
it reaches the shape of a semicircle between the two locking points, and, after this
stage, the line will star to spiral around the two locking points, until a critical
stage is reached when the two parts of the line approach towards each other in opposite
directions. As the lines move in opposite directions (their Burgers vector are equal
but of opposite sign), the dislocation will disappear as they meet (lines with Burgers
vectors of opposite signs cancel out), resulting in a closed loop dislocation which
will continue to grow outwards under the influence of the applied stress. At the same
time, the original dislocation line between the two locking points has been regenerated
and is now free two repeat the whole process.

\section{Dislocation Obstacles}

If an obstacle occurs on a slip plane, the dislocation generated from a Frank-Read
source will tend to pile up against it (Figure~\ref{fig:disloc_pile-up}).
\begin{figure}[h]
   \center{\includegraphics{disloc_pile-up.eps}}
   \caption{Model of dislocation pile-up.}
   \label{fig:disloc_pile-up}
\end{figure}
The most recent dislocations will exert a force on the older ones, so that dislocations
nearer to the the obstacle will be much closely spaced that those nearer to the source.
The pile-up exerts a back stress $\tau_\mr{b}$ on the dislocation source, which will
continue to produce dislocations until $\tau_\mr{b}$ equals the applied stress $\tau$
less the stress $\tau_0$ that activates the source:
\begin{eqnarray}
   \tau_\mr{b} = \tau - \tau_0
   \label{eq:back_stress}
\end{eqnarray}
The number of dislocation which can be piled into length $l_0$ of a slip plane in which
a shear stress $\tau$ acts cab be found as follows:
\begin{eqnarray}
   n = \frac{\pi l_0 \tau K}{G b}
   \label{eq:pile-up}
\end{eqnarray}
where $K = 1$ for edge dislocations, and $K = 1 - \nu$ for screw dislocation.
