21-Mat-A5 Phase Transformations and Thermal Treatment · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2017 — 10-Met-A5, Mechanical Behaviour and Fracture of Materials. Three hours, closed book, any Casio- or Sharp-approved calculator permitted. Eight questions of 20 marks each; the rubric states that five questions constitute a complete paper and that only the first five appearing in the answer book are marked. All eight are answered here, because this set is a study resource rather than an exam script. Several sub-parts explicitly call for an essay-format answer, and the rubric rewards clarity and organisation, so those answers are written as structured prose rather than as note form.
Nothing on the paper is a phase-transformation or heat-treatment question in the TTT/CCT, hardenability or tempering sense; the syllabus actually examined is dislocation theory, slip and twinning, strengthening mechanisms, creep, fatigue, toughness and fracture mechanics, and deformation processing.
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.
Ductility of a polycrystal depends on being able to accommodate an arbitrary shape change imposed by neighbouring grains, which by the von Mises criterion requires at least five independent slip systems operating simultaneously. At room temperature, HCP metals such as Mg, Zn and Ti-alpha slip almost exclusively on the single basal plane $(0001)$ in the $\langle11\bar20\rangle$ directions. That plane offers three slip systems, but because the three $\langle11\bar20\rangle$ directions sum to zero within the plane, only two of them are independent — far short of the five required. The non-basal systems that could supply the missing independence (prismatic $\{10\bar10\}$ and pyramidal $\{10\bar11\}$) exist, but their critical resolved shear stress at room temperature is typically an order of magnitude higher than the basal CRSS, so they do not activate under ordinary loads. Deformation twinning can partially compensate, but twinning is polar (it accommodates strain in only one sense along a given plane, per Question 2c) and cannot by itself supply general compatibility either. The result is that grain boundaries in an HCP polycrystal cannot be locally satisfied by slip alone; incompatibility stresses pile up at boundaries and triple points, nucleating cracks there well before the matrix has exhausted its ductility. FCC ($\{111\}\langle110\rangle$, 12 systems, 5 independent) and BCC (multiple $\{110\}/\{112\}/\{123\}\langle111\rangle$ families, effectively unlimited independent systems) both clear the von Mises bar easily, which is the crystallographic root of their greater room-temperature ductility.
Hall-Petch strengthening is the increase in yield strength that follows from refining the grain size of a polycrystalline metal, expressed by
$$\sigma_y = \sigma_0 + k_y\,d^{-1/2}$$where $\sigma_0$ is the friction stress opposing dislocation motion in a single crystal, $k_y$ is a material constant (the "unpinning" or Hall-Petch slope) and $d$ is the mean grain diameter. The physical mechanism is a dislocation pile-up: a dislocation source inside a grain, activated at a modest applied stress, emits dislocations onto a slip plane that terminates at the grain boundary, since the boundary's misorientation with the neighbouring grain blocks direct slip transmission. As more dislocations are emitted they pile up behind the leading one, and the stress concentration at the head of a pile-up of $n$ dislocations is amplified over the applied stress by a factor that grows with $n$, and hence with the pile-up length — which is limited by the grain diameter $d$. Slip is transmitted into the neighbouring grain (activating a source there, or nucleating a crack) once this locally amplified stress reaches a critical value $\tau_c$, independent of $d$. Solving that pile-up condition for the applied stress at which this first occurs reproduces the $d^{-1/2}$ dependence: a smaller grain permits a shorter pile-up, so a higher applied stress is needed before the local stress concentration reaches $\tau_c$. Because it is one of the very few strengthening mechanisms that improves both strength and toughness together (finer grains also block cleavage crack propagation and lower the ductile-to-brittle transition temperature), grain refinement is used wherever the specification allows it, in contrast to the strength/toughness trade-off inherent in most other mechanisms.
Both mechanisms raise the flow stress by making it harder for a dislocation to glide, but they act through geometrically different obstacles and therefore show different behaviour with composition and processing.
Solid-solution strengthening arises from individual solute atoms distributed substitutionally or interstitially throughout the lattice. Each solute atom that differs in size from the solvent produces a local elastic misfit strain field, and one that differs in shear modulus produces a local modulus-mismatch field; a gliding dislocation's own strain field interacts with these, and the dislocation must do extra work to move through the resulting spatially-fluctuating stress landscape (Fleischer/Labusch theory). Because the obstacles are single atoms distributed continuously, the strengthening increment scales smoothly with solute content — approximately as the square root of atomic fraction for a dilute random solid solution — with no maximum: more solute (up to the solubility limit) always adds more strength.
Precipitation hardening instead relies on discrete second-phase particles, spaced a distance $L$ apart, that a dislocation encounters intermittently rather than continuously. Two competing responses are possible: for small, coherent, ordered particles the dislocation can shear straight through, paying the cost of creating new particle/matrix interface, disordering an ordered structure (an antiphase boundary), or working against a coherency strain — a cost that increases with particle size as the ageing time increases. For larger, non-coherent particles the dislocation instead bypasses them by the Orowan mechanism of Question 1(a), $\tau=Gb/L$, a stress that decreases as the particles coarsen and $L$ grows. These two mechanisms cross over at a critical particle size, producing the characteristic strength maximum at peak aging — the strength first rises with ageing time as shearable particles grow (shearing gets harder), then falls as the particles coarsen past the crossover and bypass becomes easier (over-ageing).
The distinguishing signature, then, is that solid-solution strengthening increases monotonically with solute content and shows no processing-time optimum, while precipitation hardening is non-monotonic in ageing time/particle size, with a distinct peak set by the shearing/bypass crossover.
