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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_theory} 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{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}
\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}
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} \label{sec:disloc_motion}

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 dislocations 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
   because 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 $x_\mr{D}$, a relationship for the macroscopic plastic shear
strain ($\gamma$) can be established:
\begin{eqnarray}
   \gamma  = b\rho_\mr{MD}x_\mr{D}
   \label{eq:disloc_strain}
\end{eqnarray}
where $\rho_\mr{MD}$ is the density of mobile dislocations. The shear strain rate
($\dot\gamma$) can be calculated from the following equation:
\begin{eqnarray}
   \dot\gamma  = b\rho_\mr{MD}V_\mr{D}
   \label{eq:disloc_strain_rate}
\end{eqnarray}
where $V_\mr{D}$ is the average dislocation velocity.

\section{Multiplication of Dislocations} \label{sec:multipl_disloc}

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, as $\rho_\mr{D}$ increases rapidly with plastic deformation
(after large amounts of deformation, $\rho_\mr{D}$ is in the range
$10^8$--$10^9$~mm$^{-2}$), there must be one or more processes by which dislocations are
generated due to plastic deformation. These processes are briefly described next:
\begin{description}

   \item [Frank-Read source] If a large enough stress is applied to the slip plane of a
   dislocation line which is locked at one or two points, a regenerative
   process of the dislocation line will take place. If locked at just one point, the
   dislocation line will spiral around the locking point, growing constantly in length; if
   locked at two points, the line will spiral around the two locking points, leading to
   the generation of closed loops that will keep generating and growing around the
   two locking points.

   \item [Multiple cross glide] If a dislocation loop is expanding on a slip plane, it is
   possible that a part of the loop that is purely screw in character slips onto another
   plane that crosses the slip plane of the expanding loop (glide of screw dislocations is
   not confined to an unique plane), so that, if the conditions are favourable,  a new
   dislocation loop can start and glide onto a plane that is parallel to the first slip
   plane. Thus, it is possible for a single dislocation loop to expand and multiply onto
   many parallel slip planes.

   \item [Multiplication by climb] A regenerative process, known as the Barden-Herring
   source, can occur by climb in a similar way to the Frank-Read mechanism.

   \item [Grain boundary sources] An important source of dislocations in poly-crystalline
   materials is the emission of dislocations from the boundary region between grains,
   which may be produced by several mechanisms.

\end{description}


\section{Dislocation Obstacles} \label{sec:disloc_obstacles}

As stated previously in Section~\ref{sec:disloc_motion} and according to the relationship
hold by Equation~\ref{eq:disloc_strain}, plastic deformation of metals is a result of the
displacement of dislocations within the crystal. Therefore, the ability of a metal to
deform depends on the ability of dislocations to move, and, hence, the flow stress is the
stress required to overcome the opposition to the movement of dislocations.

In an otherwise perfect crystal, the only resistance to the motion of a dislocation
is the overcoming of the Peierls stress, which arises from the periodic variation of
energy in the lattice; when a dislocation moves out of a lattice equilibrium position
(where its energy is a minimum), the energy of the dislocation increases, and its
maximum value (Peierls energy) is reached half way between two equilibrium positions.
However, in real crystals, other obstacles oppose the motion of dislocations, such as
other dislocations, grain boundary, and impurity atoms. Since work hardening results
mainly from the hindering effect of other dislocations as they are generated as a
result of plastic deformation (Section~\ref{sec:multipl_disloc}), they are thought to
be the most relevant type of obstacle opposing dislocation motion.

As referred in Section~\ref{sec:multipl_disloc}, since all metals contain a network
of dislocations, it follows that every slip plane is crossed by other dislocations,
and, as a dislocation moves on its slip plane, it will have to intercept the
dislocations crossing the slip plane. The latter are called forest dislocations. As
plastic deformation proceeds, $\rho_\mr{D}$ is increased and, therefore, also the
number of forest dislocations. Moreover, the intersection of one dislocation with
another dislocation produces steps, called jogs, on each dislocation line, whose
length is equal to the Burgers vector of the dislocation that has intercepted it.
Jogs in a pure edge dislocation possess screw character, and thus do not affect the
subsequent glide of the dislocation. However, since jogs on pure screw dislocations
have edge character, the dislocation can only glide if the slip plane of the
dislocation is the same of that of the jog; otherwise, the dislocation can only move
forward and take the jog with it by climb, whose process requires thermal activation,
and, consequently, the movement of the screw dislocation will be temperature
dependent.

Generally, the dynamics of a dislocation attempting to overcome an obstacle can be
described as follows: when a dislocation approaches an obstacle, the obstacle will
impinge a force opposing the motion of the dislocation ($F_\mr{B}$), which will
increase in strength as the dislocation gets nearer to the obstacle, up to a point
where the opposing force reaches a maximum value ($F_\mr{B}^\mr{max}$), and, as the
dislocation surpasses the obstacle, this force rapidly decreases. If the length of the
obstacle is $l_\mr{B}$, the applied force of the dislocation on the obstacle will be
$\tau b l_\mr{B}$, where $\tau$ is the applied stress on the slip plane (thermal and
damping effects are neglected, so that $F_\mr{B}$ is the only force acting on the
dislocation). As a result,
\begin{eqnarray}
   \tau \geq \frac{F_\mr{B}^\mr{max}}{bl_\mr{B}}
   \label{eq:disloc_obstacle}
\end{eqnarray}
so that the dislocation can surmount the obstacle.

\section{Flow Stress} \label{sec:disloc_flow_stress}

As mentioned in Section~\ref{sec:disloc_obstacles}, the flow stress is the stress required
to overcome the force exerted by obstacles to the motion of dislocations. In this section,
the effects of temperature and strain rate on the flow stress of metals are analysed. As
shown in Figure~\ref{fig:flow_stress_regions},
\begin{figure}[h]
   \center{\includegraphics{flow_stress_regions.eps}}
   \caption[Variation of flow stress with temperature and strain rate.]
   {Variation of flow stress with temperature and strain rate~\cite{kumar68,kumar69}.}
   \label{fig:flow_stress_regions}
\end{figure}depending on the value of strain rate and
temperature, four different regions can be distinguished~\cite{kumar68,kumar69}:
\begin{description}

   \item [Thermally activated (region I)]  In this region, both strain rate and temperature
   are relatively low, and if a dislocation gliding on its slip plane encounters an
   obstacle, it will be momentarily arrested until the combined action of thermal fluctuations and the
   applied stress assist the dislocation surmount the obstacle.
   Since, by increasing the strain rate, the time a dislocation is arrested by an obstacle
   is reduced, the effects of thermal fluctuations will also be reduced, and, consequently, a
   higher stress is required to continue motion. Therefore, the flow stress decreases with
   increasing temperature, and it increases with increasing strain rate.

   \item [Athermal (region II)] In region~I, the applied stress required for the material
   to flow is divided into an thermally activated ($\tau^*$) and an athermal
   ($\tau_\mr{A}$) stress component:
   \begin{eqnarray}
      \tau = \tau^* + \tau_\mr{A}
      \label{eq:disloc_stress_1}
   \end{eqnarray}
   When the temperature increases, only $\tau^*$ is reduced (and consequently $\tau$),
   down to the point that $\tau^* = 0$, and thus $\tau = \tau_\mr{A}$, so that any further
   increase in temperature will not aid dislocations to surmount obstacles. Also, since
   the time arrested by a dislocation at an obstacle becomes irrelevant, the flow stress
   becomes also independent of strain rate.

   \item [Diffusion controlled (region III)] When the temperature becomes higher than
   $0.5T_\mr{M}$, where $T_\mr{M}$ is the melting temperature in degrees K, diffusion
   controlled mechanisms become significant. Edge dislocation and previously immobile jogs
   on screw dislocations can now climb. Since these mechanisms are thermally activated, the
   flow stress decreases rapidly with increasing temperature, and since, at higher strain
   rates, there is less time for diffusion to aid the process, flow stress increases with
   increasing strain rate.

   \item [Viscous damping (region IV)] At high dislocation speeds, a damping force
   opposing the movement of dislocations becomes relevant. This force is proportional
   to the dislocation speed $V_\mr{D}$. Furthermore, at very high strain rates, the
   applied stress is high enough to overcome instantaneously the dislocation
   obstacles without the aid of thermal fluctuations. In this region, a linear
   relationship between the applied stress and strain rate is observed:
   \begin{eqnarray} \tau = \tau_\mr{B} + m\dot\gamma \label{eq:disloc_stress_2}
   \end{eqnarray} where $\tau_\mr{B}$ is the stress needed to overcome obstacles like
   forest dislocations, and $m$ is the slope of the relationship between $\tau$ and
   $\dot\gamma$ above $\tau_\mr{B}$. Moreover, stress $\tau_\mr{B} = \tau_0^* +
   \tau_\mr{A}$, where $\tau_0^*$ is the value of $\tau^*$ at 0~K, and the stress
   attributed to damping $\tau_\mr{D} = m\dot\gamma$.
\end{description}
