\section{\Large Question Sheet}

Please use this question sheet as a supplement to the on-line edX sets.

\subsection{Martensite}
\renewcommand{\descriptionlabel}[1]{%
  \hspace\labelsep \upshape\bfseries #1%
}
\begin{description}

\item{a. } Give two examples  of deformations that change the crystal structure and two examples of those which do not.  \hfill{[20\%]}

\item{b. }  A simple shear can transform the Cubic~F structure of austenite into the hexagonal close-packed $\varepsilon$ martensite. This consists of a displacement of $\frac{a}{6}\langle 112\rangle$ on successive close-packed planes. Calculate the shear strain associated with the transformation. ($a$ is the lattice parameter of austenite).
 \hfill{[20\%]}

\item{c. } How many different crystallographic variants of $\varepsilon$-martensite can in principle form within a single austenite grain? \hfill [20\%]

\item{d. } Fig.~\ref{fig:stereogram} shows a stereographic projection of austenite showing the $\varepsilon$ habit planes of the form $\{111\}$ and shear directions of the form $\langle 112\rangle$. Which is the most favoured crystallographic variant that is stimulated to grow by applying a tensile stress along the direction [123]?\hfill{[30\%]}

\item{e. } Which alloying element promotes the hexagonal close-packed form of iron, and hence $\varepsilon$-iron, under ambient conditions? \hfill{[10\%]}.

\end{description}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=6cm]{stereogram.eps}
\caption{Stereographic projection of austenite, showing also the location of the tensile axis along [123].}
\label{fig:stereogram}
\end{center}
\end{figure} 


\clearpage
\subsection{Bainite and Martensite}

\begin{description}
\item{a. } Describe an experiment that can establish whether or not carbon diffuses  during the growth of bainite.  \hfill{[20\%]}

\item{b. } A particular steel has the chemical composition Fe-0.2C-2Si-2Mn\,wt\%, the $T_0$ curve of which is presented below. Beginning with the fully austenitic state, it is transformed isothermally to bainite at 400$^\circ$C until the reaction stops. Calculate the volume fractions of the phases present, and the carbon concentration of the austenite when the reaction ceases. Comment on the role of the silicon and state any assumptions involved in your calculation. \hfill{[30\%]}

\item{c. } Describe what might happen when the mixture of phases generated by the experiment above is cooled to ambient temperature. \hfill{[10\%]}.

\item{d. } In a second experiment, the steel is heated into the fully austenitic state until the austenite grain size reaches  40\,$\upmu$m. It is then quenched to produce martensite. If the free energy change accompanying martensitic transformation is 1000\,${\rm J\,mol^{-1}}$, and the shear and dilatational components of the shape deformation are $s=0.26$ and $\delta=0.03$, calculate the  maximum thickness of the martensite plates that form.\hfill{[25\%]}


\item{e. } Explain how the thickness of the bainite plates produced in (b) might compare against the calculated thickness of the martensite plates.
\hfill{[15\%]}

\end{description}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{T_0.eps}
\end{center}
\end{figure} 
[The strain energy per unit volume of an elastically accommodated plate of martensite is ${c\over r}\mu (s^2+\delta^2)$, where $\mu=80\,$GPa is the shear modulus of the austenite, and  ${c\over r}$ is the thickness to length ratio of the martensite plate. The molar volume of iron is about $7\times 10^{-6}\,{\rm m^3\,mol^{-1}}$.]
\clearpage


\clearpage
\subsection{Ferrite}

\begin{description}
\item{a. } The data given in the table below describe how the thickness of an allotriomorph of ferrite varies with time ($t$) during the isothermal transformation of austenite of grain size 20\,$\upmu$m. Plot the data, both as a function of $t$ and $\sqrt{t}$.  Explain the form of the relationship between the thickness of the ferrite and the transformation time. 
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|}
\hline
Thickness of $\alpha$ layer / $\upmu$m& Time / s \\
\hline
0&	0\\
0.47 &	2\\
0.64	& 4\\
0.90 & 	8\\
1.28	& 16\\
\hline
\end{tabular}
\end{center}
\label{default}
\end{table}%
 \hfill{[30\%]}

\item{b. } Describe what is meant by {\em hard impingement} and {\em soft impingement}. \hfill [10\%]

\item{c. } Illustrate the distribution of solute ahead of the transformation interface, marking on your diagrams the equilibrium compositions of the parent ($\gamma$) and product $(\alpha)$ phases ($c^{\gamma\alpha}$ and $c^{\alpha\gamma}$ respectively) and the average composition $\overline c$.  How would you expect the concentration profile to change as the thickness of the ferrite layer becomes a substantial proportion of the austenite grain size? 
\hfill{[30\%]}

\item{d. } In a ternary steel containing both manganese and carbon as solutes, the austenite is transformed isothermally into allotriomorphic under paraequilibrium conditions. Sketch an isothermal section of the iron-rich part of the Fe-Mn-C paraquilibrium diagram, indicating tie-lines within the $\alpha+\gamma$ phase field. Draw on this diagram the equilibrium $\alpha+\gamma$ phase field together will illustrative tie-lines.  By comparing the equilibrium and paraequilibrium ties lines, indicate the solute that is trapped as ferrite grows. Explain why the phase boundaries of these two diagrams meet when the Mn concentration is zero. \hfill{[30\%]}

\end{description}

\clearpage
\subsection{Ferrite, TRIP, Bainite}

\begin{description}
\item{a. } Why does the common tangent construction, on  plots of the free energy versus composition curves in a binary system,  lead to the equilibrium compositions of the phases where the tangent touches the curves simultaneously? At a constant temperature, the equilibrium compositions in a binary system are connected by a unique tie-line. Why is the tie-line no longer unique in the ternary case?  \hfill [30\%]

\item{b. } What is meant by {\em paraequilibrium} in the context of an Fe-Mn-C steel? Comment on how the concentration of the solutes would be expected to vary across the ferrite-austenite interface during paraequilibrium growth, assuming that you can measure the composition with atomic resolution. \hfill [20\%]

\item{c. } Explain qualitatively why the strain caused by deformation-induced martensitic transformation in TRIP-assisted steel cannot explain the remarkably ductility observed in spite of the high-strength of these alloys. What then is the cause of this useful combination of strength and ductility?\hfill [20\%]

\item{d. } Describe four conditions necessary to facilitate the production of tonnage quantities of nanostructured steel. Given an interfacial energy per unit area of 0.2\,J\,m$^{-2}$, estimate the energy stored in a material where the interfaces are spaced 20\,nm apart. \hfill [30\%]
\end{description}

\clearpage


\subsection{Pearlite}

\begin{description}
\item{a. } Why does pearlite in an Fe-C alloy grow at a constant rate, whereas the rate of growth of allotriomorphic ferrite decreases as the ferrite thickens?  \hfill{[10\%]}

\item{b. } Derive an expression relating the interlamellar spacing $S$,  and the amount of ferrite ($\alpha$)-cementite ($\theta$) and the interfacial area per unit volume $S_V$. What is the free energy consumed in creating these interfaces, as a function of $S$? If the chemical free energy change available for transformation is $2.86\times 10^7\,{\rm J\,m^{-3}}$, and the $\alpha/\theta$ interfacial energy per unit area is ${\rm 0.6\,J\,m^{-2}}$, find the critical interlamellar spacing $S_C$ whence all the driving force is consumed in creating the interfaces.
\hfill{[30\%]}

\item{c. } Show therefore, that the driving force available for transformation ($\Delta G$) after accounting for the $\alpha/\theta$ interfacial energy, can be expressed as $\Delta G \propto (1-[S_C/S])$. \hfill{[10\%]}

\item{d. } How does the gradient of concentration at the transformation front vary with the interlamellar spacing? Assuming that the growth velocity is proportional to $\Delta G$, and to the gradient, show that the maximum in growth rate occurs when $S=2S_C$.
\hfill{[30\%]}


\item{e. }  Explain the microstructural features of pearlite that control (a) its strength and (b) its toughness. \hfill{[20\%]}


\end{description}



