21-Mat-A4 Deformation Behaviour and Properties of Materials · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2017 — 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 parts of Question VI (grain-size strengthening) touch mechanical properties directly; the paper as a whole is a broad introductory materials-science survey — electron structure, bonding, crystal structure, crystallographic planes, solid solubility, XRD, diffusion 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:
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Given. The printed Fe–Fe$_3$C diagram labels: melting point of pure iron 1538°C; peritectic ($L+\delta\to\gamma$) at 1493°C; $\delta\to\gamma$ transformation at 1394°C; eutectic ($L\to\gamma+\text{Fe}_3\text{C}$) at 1147°C, eutectic point 4.30 wt% C; maximum C solubility in austenite 2.14 wt% at 1147°C; $\gamma\to\alpha$ transformation at 912°C; eutectoid ($\gamma\to\alpha+\text{Fe}_3\text{C}$) at 727°C, eutectoid point 0.76 wt% C; maximum C solubility in $\alpha$-ferrite 0.022 wt% at 727°C; cementite (Fe$_3$C) fixed composition 6.70 wt% C.
[Figure not reproduced: Fig. VII — Fe–Fe$_3$C phase diagram reconstructed from the labelled points and boundaries printed on the exam figure, with the two queried state points marked. See the official exam paper.]
The relevant boundary at 1000°C is the austenite solvus (the "Acm" line), which runs from 2.14 wt% C at 1147°C down to the eutectoid 0.76 wt% C at 727°C. Interpolating along this curved boundary (concave toward the eutectoid, as drawn on the diagram) places the Acm line at roughly 1.3–1.5 wt% C at 1000°C. Since the queried composition, 2 wt% C, exceeds this boundary, the alloy at this state point lies to the right of the Acm line, in the two-phase field: $$\boxed{\text{Phases present: } \gamma\ \text{(austenite)} + \text{Fe}_3\text{C}\ \text{(cementite)}}$$
Both maxima are printed directly on the diagram at the temperature where each single-phase field is widest:
| Phase | Maximum C solubility | At temperature |
|---|---|---|
| $\alpha$-ferrite (BCC) | 0.022 wt% C | 727°C (eutectoid temperature) |
| $\gamma$-austenite (FCC) | 2.14 wt% C | 1147°C (eutectic temperature) |
Austenite (FCC) dissolves roughly 97× more carbon than ferrite (BCC) at its respective maximum, because the FCC octahedral interstitial site ($r=0.052$ nm) is substantially larger than the BCC tetrahedral site ($r=0.036$ nm) that hosts interstitial carbon in ferrite — this is the structural basis for quench-hardening steel (trapping FCC-dissolved carbon into a supersaturated, distorted BCT martensite on rapid cooling).
Increasing carbon content raises both the volume fraction and the interlamellar fineness of the hard, brittle cementite (Fe$_3$C) phase relative to the soft, ductile ferrite matrix (via the lever rule at any temperature below the eutectoid). Strength and hardness increase steadily with carbon content, from soft, ductile pure ferrite ($\sim0$ wt% C) through increasingly pearlitic hypoeutectoid steels to the fully pearlitic eutectoid composition (0.76 wt% C) and on into hypereutectoid steels, where hardness continues to rise (more cementite). Ductility falls correspondingly: cementite itself is essentially non-ductile, and beyond the eutectoid composition, proeutectoid cementite begins forming as a continuous network along the prior-austenite grain boundaries, which is especially damaging to ductility and impact toughness because it provides a continuous, low-energy crack path. The practical consequence is the classic engineering trade-off in plain-carbon steel selection: higher-carbon steels are stronger/harder but must be used at correspondingly lower ductility and toughness.
Given. Alloy composition $C_0=0.35$ wt% C (hypoeutectoid, since $0.35<0.76$ eutectoid); just below 727°C the two phases present are $\alpha$-ferrite ($C_\alpha=0.022$ wt%) and Fe$_3$C ($C_{Fe_3C}=6.70$ wt%, the tie-line endpoints immediately below the eutectoid isotherm).
Find. The mass fractions $W_\alpha$ and $W_{Fe_3C}$.
Approach. Apply the lever rule across the $\alpha+\text{Fe}_3\text{C}$ tie line at this temperature, using the two phase-boundary compositions as the tie-line endpoints.
| Phase | Mass fraction |
|---|---|
| $\alpha$-ferrite | 95.09% |
| Fe$_3$C (cementite) | 4.91% |
A eutectic reaction is the isothermal, invariant transformation of a single liquid phase directly into two distinct solid phases on cooling (and the reverse on heating): $\text{liquid}\rightleftharpoons\text{solid}_1+\text{solid}_2$. A eutectoid reaction is the analogous isothermal, invariant transformation of a single solid phase into two different solid phases: $\text{solid}_1\rightleftharpoons\text{solid}_2+\text{solid}_3$. For the Fe–C system specifically:
| Reaction | Equation | Temperature | Composition |
|---|---|---|---|
| Eutectic | $L\rightleftharpoons\gamma+\text{Fe}_3\text{C}$ | 1147°C | 4.30 wt% C |
| Eutectoid | $\gamma\rightleftharpoons\alpha+\text{Fe}_3\text{C}$ | 727°C | 0.76 wt% C |
