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.
A standard tensile test lasts minutes and applies its load once, so it cannot expose either time-dependent or cycle-dependent damage; both creep and fatigue need conditions the tensile test never supplies.
(i) Creep requires a stress sustained continuously at a temperature high enough, relative to the metal's melting point, for thermally activated processes — dislocation climb, or vacancy diffusion to and from grain boundaries — to contribute significantly to the strain rate on the timescale of interest. The controlling variable is the homologous temperature $T/T_m$, not the absolute temperature: creep becomes engineering-significant above roughly $0.3$-$0.4\,T_m$, which is why lead creeps on a bench at room temperature but a structural steel does not become a creep concern until several hundred degrees Celsius. Given enough time at such a temperature, even a stress well below $\sigma_y$ produces a finite, and eventually unbounded, plastic strain.
(ii) Fatigue requires a fluctuating stress with a tensile component, repeated for a sufficient number of cycles, acting at (or producing) a local stress concentration — a free surface, notch, inclusion or weld toe. No elevated temperature is required. Because the damage is local (confined to persistent slip bands in a few favourably oriented surface grains, Question 6c), a nominal stress far below $\sigma_y$ can still drive local plastic strain at those sites cycle after cycle, eventually nucleating and growing a crack to failure even though the bulk material never yields.
At intermediate stress and temperature (below the diffusional-creep regime but above the athermal glide regime), the dominant mechanism is dislocation climb-controlled creep. Dislocations glide on their slip planes until they encounter obstacles — other dislocations, precipitates — that they cannot get past by glide alone at the applied (sub-yield) stress. At elevated temperature, however, a dislocation held up at such an obstacle can absorb or emit vacancies at its core and climb a short distance out of its original slip plane, onto a parallel plane where the obstacle no longer blocks it, and then resume glide. This glide-climb-glide sequence lets deformation continue steadily at a stress that would produce no further strain at all without the thermally activated climb step, which is exactly why the process is impossible below the temperature at which vacancy diffusion becomes fast enough (again, roughly $0.3$-$0.4\,T_m$).
The rate-controlling step is the climb event, so the steady-state (secondary-stage) creep rate follows an Arrhenius, power-law form:
$$\dot\varepsilon_s = A\,\sigma^n\exp\!\left(-\frac{Q_c}{RT}\right)$$where $Q_c$ is close to the activation energy for self-diffusion (confirming that vacancy-mediated climb is rate-controlling) and $n$ is typically 3-8, distinctly different from the $n=1$ (linear-in-stress) signature of the diffusional (Nabarro-Herring/Coble) creep mechanism described for Question 6(b) — the stress exponent is the diagnostic that distinguishes which mechanism is operating at a given stress/temperature combination. Because climb is thermally activated and depends on dislocation density evolving with strain (recovery competing with work hardening), the resulting creep curve shows the classic primary (decelerating, as dislocation density and internal back-stress build), secondary (steady-state, recovery balancing hardening) and tertiary (accelerating, as internal damage — necking or, more commonly at this stress/temperature range, cavitation as in Question 6b — reduces the load-bearing section) stages sketched in Question 4(a).
The argument is a competition between two stresses that respond to strain rate in opposite ways. The yield stress of a metal whose plasticity is limited by lattice (Peierls-Nabarro) friction — the case for BCC metals, where screw-dislocation motion requires thermally activated kink-pair nucleation — rises sharply as strain rate increases, because a higher rate leaves less time for the thermal activation that lowers the effective barrier:
$$\sigma_y=\sigma_y^*(T,\dot\varepsilon)+\sigma_G$$with only the athermal component $\sigma_G$ (from solutes, precipitates, boundaries) independent of rate. The cleavage fracture stress $\sigma_f$, by contrast, is set by a Griffith-type condition for propagating a sharp microcrack through the lattice, $\sigma_f\approx\sqrt{2E\gamma_{\text{eff}}/\pi c}$, and depends on essentially rate-independent quantities (modulus, surface energy, flaw size), so it stays nearly flat as strain rate rises.
Plotted against strain rate, the two curves cross: below the crossover rate the material still yields before it can cleave, blunting any incipient microcrack and failing by ductile void growth; above it, $\sigma_y$ has risen past $\sigma_f$, so the very first microcrack nucleated at general yield propagates immediately, and the material fractures with almost no plastic work at a stress at or below what would otherwise have been its yield point. This is the same physical crossover that produces the ductile-to-brittle transition temperature, and it is why increasing loading rate raises the effective DBTT — a steel that behaves ductilely in a slow bend test can fracture in a brittle manner in an impact (Charpy) test at the identical nominal temperature. Materials without a strongly rate-dependent yield stress — FCC metals such as austenitic stainless steel, copper and aluminium, whose Peierls stress is negligible — never see $\sigma_y$ overtake $\sigma_f$ at any accessible rate, and consequently show no ductile-to-brittle transition and no comparable rate-driven brittleness, which is exactly why austenitic stainless steel is specified over ferritic steel for cryogenic and impact-loaded service at equivalent nominal strength.
