21-Mat-A4 Deformation Behaviour and Properties of Materials · December 2018
Question 7 of 7: Question VII: Phase Diagram
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, December 2018 — 10-Met-A4, Structure of Materials. Three hours, closed book, one approved calculator (Casio or Sharp). Seven questions of 20 marks each (Roman numerals I–VII); the rubric asks for any five, with only the first five in the answer book marked. All seven are solved here, because this set is a study resource rather than an exam script. All necessary equations, constants and an error-function table are provided in the exam's own appendix (reproduced where used below).
Note on the exam title. The printed exam header reads 10-Met-A4, Structure of Materials. Only Question VI.2(c) (grain-size strengthening) touches mechanical/deformation properties directly; the paper as a whole is a broad introductory materials-science survey — atomic structure, bonding, crystal structure/directions/planes, point defects, XRD, dislocations and phase diagrams — and is answered as such below.
Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:
W. D. Callister and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed. — atomic structure and bonding (Ch. 2), crystal structure and Miller/Miller–Bravais indices (Ch. 3), imperfections/point defects (Ch. 4), XRD (Ch. 3), dislocations/slip/Hall–Petch (Ch. 4, 7), phase diagrams (Ch. 9).
G. E. Dieter, Mechanical Metallurgy, 3rd ed. — grain-boundary strengthening and dislocation theory.
Given. The printed Fe–Fe$_3$C diagram labels: eutectic ($L\to\gamma+\text{Fe}_3\text{C}$) at 1147°C, 4.30 wt% C; maximum C solubility in austenite 2.14 wt% at 1147°C; eutectoid ($\gamma\to\alpha+\text{Fe}_3\text{C}$) at 727°C, 0.76 wt% C; maximum C solubility in $\alpha$-ferrite 0.022 wt% at 727°C; peritectic ($\delta+L\to\gamma$) at 1493°C; cementite (Fe$_3$C) fixed at 6.70 wt% C; the cooling path is drawn at the alloy's own composition, 1.5 wt% C, with point x at 1100°C (labelled in the $\gamma$ field), point y$'$ just above 727°C, and point z below 727°C.
[Figure not reproduced: Fig. VII — Fe–Fe$_3$C phase diagram reconstructed from the labelled points on the exam figure, with the alloy's 1.5 wt% C cooling path and the three queried state points x, y$'$, z marked. See the official exam paper.]
VII.1 — Invariant Points, Eutectic and Eutectoid Reactions
Invariant reactions on the Fe–Fe$_3$C diagram
Reaction
Temperature
Composition
Reaction equation
Peritectic
1493°C
$\delta$ (0.09%C) + L (0.53%C) → $\gamma$ (0.17%C)
The eutectic (4.30%/1147°C) and eutectoid (0.76%/727°C, plus the 0.022% ferrite solubility limit) compositions are explicitly printed on this exam's own figure. The peritectic point's three compositions (0.09/0.17/0.53 wt% C) are not printed on this particular exam figure — they are the standard textbook Fe–Fe$_3$C values (Callister), included above for completeness since the question asks generically "where are the invariant points."
VII.2 — Phases at Point x (1.5 wt% C, 1100°C)
The exam's own figure places point x inside the $\gamma$ (austenite) single-phase field at 1100°C: this temperature lies below the eutectic isotherm (1147°C) but the alloy's 1.5 wt% C composition is still below the $\gamma/(\gamma+\text{Fe}_3\text{C})$ solvus (Acm line) at 1100°C, which runs from 2.14 wt% C at 1147°C down to 0.76 wt% C at 727°C (comfortably above 1.5 wt% C at 1100°C).
$$\boxed{\text{Point x: single phase } \gamma\ \text{(austenite), composition} = 1.5\ \text{wt\% C}}$$
Since only one phase is present, its composition must equal the bulk alloy composition — there is no second phase to partition carbon into.
VII.3 — Microstructure Evolution Along the Cooling Path
Fig. VII.3 — schematic microstructures at the three queried points along the 1.5 wt% C cooling path (blue = austenite grains/pearlite lamellae, red outline = proeutectoid cementite).
The alloy is hypereutectoid (1.5 wt% C $>$ the 0.76 wt% eutectoid composition), which fixes the sequence of microstructures below:
At x (1100°C): single-phase $\gamma$ (austenite) — uniform, equiaxed grains, composition 1.5 wt% C throughout (Question VII.2).
At y$'$ (just above 727°C): as the alloy cools through the $\gamma+\text{Fe}_3\text{C}$ two-phase field, proeutectoid ("primary") cementite nucleates and grows preferentially along the prior-austenite grain boundaries, forming a continuous network; the remaining $\gamma$ is progressively depleted in carbon, approaching the eutectoid composition (0.76 wt% C) as $T\to727^{\circ}\text{C}$.
At z (below 727°C, sketched at 600°C): at 727°C the remaining $\gamma$ (now exactly at the eutectoid composition) transforms by the eutectoid reaction into pearlite (alternating fine lamellae of $\alpha$-ferrite and Fe$_3$C); the proeutectoid Fe$_3$C grain-boundary network formed above 727°C does not itself transform further and persists unchanged. Final microstructure: proeutectoid Fe$_3$C network + pearlite matrix (quantified in Question VII.4).
VII.4 — Phase Fractions at Point z: Pearlite and Proeutectoid Cementite
Given. $C_0=1.5$ wt% C (hypereutectoid); at the eutectoid isotherm the tie-line endpoints are $\gamma=0.76$ wt% C (eutectoid composition) and Fe$_3$C $=6.70$ wt% C.
Find. The mass fractions of proeutectoid Fe$_3$C and of pearlite in the final (room-temperature-equivalent) microstructure at point z.
Approach. Apply the lever rule on the $\gamma+\text{Fe}_3\text{C}$ tie line just above 727°C to get the proeutectoid cementite fraction $W'_{Fe_3C}$; the remaining austenite fraction transforms entirely, unchanged in amount, to pearlite at the eutectoid reaction.
Pearlite fraction. All of the remaining (untransformed) austenite becomes pearlite at the eutectoid reaction, so:
$$W_{pearlite}=1-W'_{Fe_3C}=1-0.1246=\boxed{87.5\%}$$
