
\chapter{Deformation and Texture}
\section{Slip in a Single-Crystal}

In a tensile test on the sample of cross-sectional area $A$ illustrated in Fig.~\ref{fig:single}a, the normal to the slip plane lies at an angle $\phi$ to the force $F$, and the slip direction is at an angle $\lambda$ to the tensile axis. The area of the slip plane is $A/\cos \phi$ and the force in the slip direction is $F \cos\lambda$ so that the shear stress on the slip plane in the slip direction is
\begin{equation}
\tau = \frac{F}{A}\underbrace{\cos\phi \cos\lambda}_{\hbox{Schmid factor}}
\end{equation}
The Schmid factor  has a maximum value of 0.5 when $\phi=\lambda=45^\circ$. Suppose that a tensile axis is along [213], plotted on the shaded stereographic triangle in Fig.~\ref{fig:single}b. There is a simple way of determining which slip system will have the largest resolved shear stress. The slip system will be defined by a slip plane and a slip direction with the latter lying in the former.  This means that the two components of the slip system must come from different stereographic triangles.
 The two stereographic triangles that are adjacent to the shaded triangle satisfy this condition. Any other combination gives angles that are too large, giving Schmid factors that are smaller. The slip system with the highest resolved shear stress when the tensile lies in the shaded triangle will therefore be [011]$(1\overline 11)$. The construction illustrated is known as Diehl's rule.


\begin{figure}[htp]
\centering
\subfigure(a){
\includegraphics[width=0.4\textwidth]{Deformation_Texture/single.eps}}
\subfigure(b){
\includegraphics[width=0.5\textwidth]{Deformation_Texture/Diehl2.eps}}
\caption{(a) Deformation of a single crystal. The force is resolved on to the slip plane and towards the slip direction. Dividing by the area of the slip plane gives the shear stress along the slip direction. (b) There are 24 stereographic triangles corresponding to the 24 slip systems of the type $\{111\}\langle1\overline 10\rangle$. The orientation of a tensile axis is plotted in the shaded shaded triangle.}  \label{fig:single}
\end{figure}

The unconstrained shear deformation of a single crystal as illustrated in Fig.~\ref{fig:rotate} leads to the rotation of the axis along which the force is applied. The normal to the slip plane, the tensile axis and the slip direction are all in the same plane and remain so during rotation. The tensile axis rotates towards the slip direction, on this common plane, during the course of deformation. If the axis along which the force is applied is not allowed to change orientation, then it is the slip system which rotates during deformation Fig.~\ref{fig:rotate}.


\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-5cm}{\includegraphics[width=0.98\linewidth]{Deformation_Texture/shear.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
\caption{Slip of an unconstrained single-crystal. Notice how the axis along which the force is applied rotates as a consequence of the slip. The amount of shear implemented is written $\alpha${\bf s}. If the force axis is constrained to be vertical then the sample will rotate relative to the testing machine by the extent illustrated.}
\label{fig:rotate}
    \end{minipage}
  \end{center} \end{figure} 

\section{Texture}

Imagine that there is a polycrystalline substance in which the individual crystals are randomly oriented relative to an {\em external frame of reference}\symbolfootnote[3]{This frame may, for example, consist of a set of vectors parallel to the rolling direction, transverse direction and thickness of a rolled plate; alternatively, the principal axes of an applied system of stresses.}. Suppose that this polycrystalline substance is subjected to rolling deformation. The individual crystals will then tend to rotate such that the slip planes comply with the external forces as illustrated in Fig.~\ref{fig:rotate}. This means that the original random distribution of crystals in the undeformed sample becomes non-random. The deformed material is said to become {\em crystallographically textured}.

Most polycrystalline materials show texture due to processing. Texture can arise during solidification when those crystals which have a fast growth direction parallel to that of the heat flow will dominate the final structure. It can arise during recrystallisation and phase transformation when selective nucleation leads to the formation of a biased distribution of crystals.

A convenient method for communicating texture caused by deformation, for example, in rolled
sheet, is by stating the set $\{h~k~l\}\langle u~v~w\rangle$, the planes which lie roughly parallel to
the rolling plane, and  direction in the rolling plane that tends to be parallel to the rolling direction. The
overall texture can be represented as the sum of components: 
\begin{equation}\hbox{texture}
=\sum_i \lambda_i \{h~k~l\}_i\langle u~v~w\rangle_i
\end{equation}
where $\lambda$ represents the weighting given to a particular type of
texture. The major components of the deformation texture of austenite are
$\{1~1~0\}\langle1~\overline 1~2\rangle$ and $\{1~1~2\}\langle1~1~\overline 1\rangle$, the so-called {\em brass} and {\em copper} textures respectively. Recrystallisation in a ferritic steel often leads to a {\em cube} texture $\{1~0~0\}\langle 0~0~1\rangle$ {\em cube} component and special thermomechanical processing of the type used to produce magnetically soft metal is associated with the {\em Goss texture} $\{1~1~0\}\langle 0~0~1\rangle$.

\begin{figure}
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Deformation_Texture/stereo1.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Deformation_Texture/stereo2.eps}}\\
\caption{(a) Figure showing a random distribution of 100 poles from a polycrystalline sample containing 500 crystals, plotted relative to the rolling direction ($RD$) and transverse direction ($TD$). Notice that the distribution of poles does not appear random but this is an artefact due to the angular distortion on a stereographic projection. (b) Corresponding pole figure after texture is introduced by phase transformation. Here there is a strong tendency for the 100 poles to cluster around the sample axes. The distribution of poles can be plotted as contours rather than the dots illustrated. }
\label{fig:pole}
\end{figure}

Texture can be plotted on a stereogram. A {\em pole figure} in this context consists of a stereogram with its axes defined relative to the external frame of reference, and with particular $hkl$ poles from each of the crystals in the polycrystalline aggregate plotted on to it. If the distribution of crystals is random, the pole figure would appear as in Fig.~\ref{fig:pole}a, and if the distribution is non-random then this would be apparent in the pole figure, as shown in Fig.~\ref{fig:pole}b.

An {\em inverse pole figure} is one in which the sample frame is plotted relative to the crystal axes of a reference crystal. Fig.~\ref{fig:inverse} an example where the rolling, normal and transverse directions (RD, ND and TD respectively) are plotted relative to the crystal axes as defined by the stereographic triangle. In a cubic system, the 24 stereographic triangles with $\langle100\rangle$, $\langle101\rangle$, and $\langle111\rangle$ at the corner of each triangle, are crystallographically equivalent. Therefore, the orientation of the rolling direction, for example, from each of the 24 triangles can be plotted on just one of the stereographic triangles in the form of poles, or contour plots representing the design of poles in angular space.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{Deformation_Texture/inverse.eps}
\caption{Inverse pole figures for a cubic crystal. The diagrams at the top have poles corresponding to the rolling, normal and transverse directions plotted relative to the crystal axes, whereas those at the bottom have the same data represented as contours representing the density of poles.}
\label{fig:inverse}
\end{center}
\end{figure} 

Crystallographic texture can be measured using a variety of techniques such as X-ray diffraction or electron back-scattered diffraction (EBSD). In the former case, the polycrystalline sample is exposed in a system set to detect X-rays from a particular reflection (Bragg angle) and the sample is systematically tilted and rotated in order to capture intensity at a variety of orientations. In EBSD, an electron beam is rocked about a fixed point on the surface of a grain using a scanning electron microscope. At particular angles, the beam is Bragg diffracted, resulting in a reduction in intensity picked up by the detector, leading to the formation of channelling patterns of the type illustrated in Fig.~\ref{fig:chanel}. These patterns can be interpreted to determine the orientation of the crystal relative to the sample axes.
\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.42\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{Deformation_Texture/chanel.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.55\linewidth}
      \caption{ An electron channelling pattern from a crystal of copper plotted on a standard stereographic triangle.}  \label{fig:chanel}
    \end{minipage}
  \end{center} \end{figure} 
 
\section{Orientation Distribution Functions}

A pole figure does not give information about the orientation of a particular crystal relative to another, since the poles are not in the final plot identified with particular crystals. An orientation distribution function is a more rigorous method of expressing the orientations of crystals relative to the frame of reference. It uses three Euler angles ($\varphi_1$, $\Phi$ and $\varphi_2$) which define the relative orientation of the crystal and sample frames of reference, as illustrated in  Fig.~\ref{fig:Euler_angles}. 


\begin{figure}[htp]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Deformation_Texture/Euler_angles1.eps}}\hfill
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Deformation_Texture/Euler_angles2b.eps}}\\[10pt]
\subfigure(c){
\includegraphics[width=0.45\textwidth]{Deformation_Texture/Euler_angles3b.eps}}\hfill
\subfigure(d){
\includegraphics[width=0.45\textwidth]{Deformation_Texture/Euler_angles4b.eps}}
\caption{Operations that define the three (Bunge) Euler angles. (a) Initial relationship between crystal and sample coordinates. (b) Rotation by $\varphi_1$ about ND to generate RD' in a position normal to the plane containing ND and 001. (c) Rotate by $\Phi$ about RD' to generate ND' parallel to 001. (d) Rotate by $\varphi_2$ about 001 to bring bring crystal and sample axes into coincidence. The insets show the rotations relative to an aircraft.}
\label{fig:Euler_angles}
\end{figure}

The set of measured Euler angles made by the $RD$, $TD$ and $ND$ directions for a particular crystal is plotted as a point in a cube whose axes scale with $\varphi_1$, $\Phi$ and $\varphi_2$. Each point inside this cube is a single crystal. 

\begin{figure}
\centering
\subfigure(a){
\includegraphics[width=0.3\textwidth]{Deformation_Texture/ODF_cube.eps}}
\subfigure(b){
\includegraphics[width=0.6\textwidth]{Deformation_Texture/ODF_sections.eps}}
\caption{(a) An orientation distribution plotted on a cube with axes defined by the Euler angles. Each dot represents a crystal orientation relative to sample axes. (b) Sections of the cube to show how the density of orientations varies.}
\label{fig:Euler}
\end{figure}


There are very many industrial processes which control and exploit texture on a grand scale. Beverage cans are incredibly thin even though their bodies are made without fracture using extreme deformation including deep drawing. The texture of the material prior to drawing is controlled such that plastic instability is avoided in the thickness direction. For similar reasons, the steels used in the manufacture of formed automobile bodies are texture-controlled. Iron alloys used to make electricity transformers have to be magnetically soft. This is achieved by controlling the crystallographic texture to minimise magnetic losses.

\clearpage\section{Summary}

Crystallographic texture can be understood by a thought experiment in which a large single crystal is broken up into many smaller ones which are then joined together in somewhat different orientations that are not random. The material then behaves as if it as some of the characteristics of a single crystal (such as anisotropy) while other properties are more a feature of  random polycrystals (such as the multiple slip systems needed per crystal in order to achieve plasticity without the creation of voids).   

We have discussed how texture can arise from plastic deformation, but any kind of processing can lead to its development. The heat treatment of deformed materials to induce recrystallisation can lead to recrystallisation textures because not all `nuclei'  grow at the same rate. The nuclei have to be sufficiently different in orientation from their surroundings to accomplish rapid growth. Similar selection can occur in phase transformations where the product phase forms at a grain boundary because certain boundaries make it possible for the product to adopt a favourable orientation relationship with the parent phase on either side of the interface. Both magnetic  and electrical fields  can also lead to orientational order in appropriate materials.

So how does one produce a polycrystalline material with a random arrangement of grains? The short answer is, with enormous difficulty. Powder techniques can partially reach this goal; metallic or ceramic powders that are sintered together tend to undergo very limited deformation or rotation during fabrication, and hence may approach a random orientational distribution. 