21-Mat-A4 Deformation Behaviour and Properties of Materials · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2013 — Met-A4, Structure of Materials. Three hours, closed book, one approved calculator (Casio or Sharp). Eight questions of 20 marks each; the rubric asks for any five, and only the first five in the answer book are marked. All eight are solved here. All necessary equations and constants are provided in the exam's own appendix (reproduced where used below).
The printed exam header reads Met-A4, Structure of Materials. Only two of the eight questions (VI and VII) are genuinely diffusion/mechanical-properties content; the paper as a whole is a broad introductory materials-science survey — bonding, crystal defects, crystallography, XRD 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.
(a) A Schottky defect is a paired vacancy in an ionic crystal: one cation and one anion (in a stoichiometric ratio that preserves local charge neutrality) both leave their lattice sites, typically migrating to the surface. It dominates in ionic solids with similar-sized cations and anions (e.g. NaCl, KCl) and, unlike a single vacancy, does not build up a net charge in the crystal. (b) A grain boundary is the two-dimensional interface between two crystallites (grains) of the same phase that share a composition but differ in crystallographic orientation; atoms in the boundary region are less densely and less regularly packed than in either grain's interior, which is why boundaries are preferential sites for diffusion, corrosion and (via Hall–Petch strengthening) obstruct dislocation motion. (c) A screw dislocation is a line defect around which the crystal planes are sheared into a helical ramp; the Burgers vector b (the closure failure of a Burgers circuit around the line) is parallel to the dislocation line itself, unlike an edge dislocation where it is perpendicular. (d) A stacking fault is a local interruption in the normal stacking sequence of close-packed planes — e.g. an FCC crystal's $\ldots$ABCABC$\ldots$ sequence briefly running as $\ldots$ABCABABCABC$\ldots$, inserting a thin region of the alternative (HCP-like) stacking. (e) A partial dislocation (Shockley partial) is a dislocation whose Burgers vector is a fraction of a full lattice translation vector; a full dislocation in an FCC metal commonly dissociates into two partials, $\mathbf{b} = \mathbf{b_1}+\mathbf{b_2}$, connected by a ribbon of stacking fault, because the dissociated configuration has lower total strain energy ($\propto b^2$) than the single full dislocation.
Given. $\Delta H_s = -3.96\ \text{kJ/mol} = -3960\ \text{J/mol}$; $R = 8.31\ \text{J/mol}\cdot\text{K}$; $S(25^{\circ}\text{C}) = S(298\ \text{K}) = 5.51\times10^{23}\ \text{atoms/(m}^3\text{}\cdot\text{atm)}$; target $T_2 = 200^{\circ}\text{C} = 473\ \text{K}$.
Find. $S(200^{\circ}\text{C})$.
Approach. Divide $S(T_2)=S_0e^{-\Delta H_s/RT_2}$ by $S(T_1)=S_0e^{-\Delta H_s/RT_1}$ to eliminate the unknown constant $S_0$.
Because $\Delta H_s$ is negative (dissolution of He into the glass network is exothermic, a weak van der Waals-type interaction rather than a chemical bond), solubility falls as temperature rises — the opposite of the more familiar activated-diffusion case, where a negative exponent multiplying a positive $Q$ makes the rate rise with $T$.
| Quantity | Value |
|---|---|
| $S(25^{\circ}\text{C})$ (given) | $5.51\times10^{23}$ atoms/(m3·atm) |
| $S(200^{\circ}\text{C})$ | $3.05\times10^{23}$ atoms/(m3·atm) |
| Yield strength vs. decreasing grain size (II.3) | increases (Hall–Petch) |
Yield strength increases as grain size decreases, following the Hall–Petch relation given in the appendix, $\sigma = \sigma_0 + kd^{-1/2}$, where $d$ is the average grain diameter. Grain boundaries are barriers to dislocation motion: a dislocation gliding on a slip plane in one grain cannot simply continue into the next grain, because the slip plane orientation changes discontinuously across the boundary and the boundary's disordered atomic structure disrupts the stress field driving the dislocation. Slip must instead re-nucleate in the neighbouring grain, which requires a local stress concentration (a pile-up of dislocations against the boundary) to trigger. A finer grain size packs more boundary area per unit volume, so pile-ups are shorter and dislocations encounter obstacles more often, raising the applied stress needed to sustain macroscopic yielding — hence smaller $d$ gives higher $\sigma$.
