21-Mat-A5 Phase Transformations and Thermal Treatment · May 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, May 2018 — 10-Met-A5, Mechanical Behaviour and Fracture of Materials. Three hours, closed book, any non-communicating 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 crystallography of slip and twinning, dislocation theory, creep, fatigue, toughness and fracture mechanics, and safe-life fatigue design.
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.
Both creep and fatigue are damage processes that accumulate with exposure, so both escape the tensile test entirely: a tensile test lasts a few minutes and applies the load once, and neither variable that matters here — time and cycles — is given a chance to act.
(i) Creep. A metal creeps when it is held under a sustained stress at a temperature high enough for diffusion to be significant on the timescale of the loading. The controlling variable is the homologous temperature $T/T_m$, not the absolute temperature: creep becomes engineering-significant above roughly $0.3$ to $0.4\,T_m$. That is why lead and solder creep on a bench at room temperature (for lead, 293 K is about $0.49\,T_m$), why aluminium creeps a little above 150 °C, and why a steel is not considered a creep problem below about 400 °C. The stress may be well below $\sigma_y$ because the deformation does not require the athermal motion of dislocations past their obstacles; thermal activation supplies the missing part of the driving force, and time supplies the rest. The three further requirements are that the stress be sustained rather than momentary, that the required service life be long compared with the time constant of the mechanism, and that the accumulated strain (or the rupture life) rather than an instantaneous strength be the design limit.
(ii) Fatigue. A metal fatigues when it is subjected to a fluctuating stress with a tensile component, repeated for a sufficient number of cycles, at a location where the local stress is concentrated — typically a free surface, a notch, a weld toe, a fretting contact or an inclusion. No elevated temperature is needed. The nominal stress amplitude may be a small fraction of $\sigma_y$ because yielding on the microscopic scale is not suppressed by a low nominal stress: in surface grains favourably oriented for slip, to-and-fro glide concentrates into persistent slip bands, and the irreversible part of that glide builds extrusions and intrusions on the free surface (Question 8c). The intrusion is a microscopic notch, and it becomes the crack. The essential conditions are therefore a cyclic (or fluctuating) load, a tensile part of the cycle to open a crack, a free surface or defect at which to start, and enough cycles — with the qualification that a mean tensile stress, a corrosive environment or a pre-existing crack all lower the threshold, and that ferrous alloys show a fatigue limit below which the crack never starts while most non-ferrous alloys do not.
Of the mechanisms available, the one that most directly answers “how can a metal deform plastically at $\sigma\lt \sigma_y$” is diffusional creep, because it involves no dislocation glide at all. It is the mechanism that dominates at low stress, high temperature and small grain size — the bottom-left region of an Ashby deformation-mechanism map.
The microstructural picture. Consider one grain of a polycrystal held under a tensile stress. The grain boundaries lying normal to the stress axis are being pulled apart; those lying parallel to it are being pushed together. A grain boundary is a sink and source for vacancies, and the equilibrium vacancy concentration at a boundary depends on the normal stress acting on it:
$$C_v \;=\; C_v^{0}\exp\!\left(\frac{\sigma_n \Omega}{kT}\right)$$where $\Omega$ is the atomic volume and $\sigma_n$ the normal stress (positive in tension). The boundaries in tension therefore hold a slightly higher equilibrium vacancy concentration than those in compression, and the difference sets up a steady vacancy flux from the transverse boundaries to the longitudinal ones. Atoms flow the other way. Material is thereby removed from the sides of the grain and plated onto its ends: the grain lengthens along the stress axis and thins across it, which is precisely a plastic strain, achieved with no slip and therefore with no need to exceed $\tau_{\text{CRSS}}$.
Two parallel paths. The vacancies can cross the grain interior, in which case the process is Nabarro–Herring creep and the rate is controlled by the lattice self-diffusion coefficient $D_L$:
$$\dot{\varepsilon}_{\text{NH}} \;=\; A_{\text{NH}}\,\frac{D_L\,\sigma\,\Omega}{d^{2}\,kT}$$or they can travel along the boundaries, a much faster path per atom but one confined to a layer of thickness $\delta$, giving Coble creep:
$$\dot{\varepsilon}_{\text{C}} \;=\; A_{\text{C}}\,\frac{\delta D_{gb}\,\sigma\,\Omega}{d^{3}\,kT}$$Both are linear in stress — a stress exponent of one, in contrast to the $n \approx 4$–$8$ of dislocation (power-law) creep — and both scale strongly with grain size, as $d^{-2}$ and $d^{-3}$ respectively. Because $Q_{gb}\approx 0.5\,Q_L$, the boundary path wins at lower temperature and the lattice path takes over as the temperature rises.
Accommodation and the end of life. The shape change of the individual grains is not by itself compatible across a polycrystal, so diffusional creep must be accommodated by grain-boundary sliding — the two are sequential parts of one process, not competitors. Where sliding is obstructed, at a triple point or at a hard second-phase particle sitting in the boundary, the accommodation fails locally and a cavity nucleates. The cavities grow by the same vacancy flux that is driving the creep, link along the transverse boundaries, and produce the characteristically intergranular, low-ductility creep rupture seen in the tertiary stage (Question 8b). That is why the practical engineering countermeasures follow directly from the equations: coarsen the grains (or remove the boundaries altogether, e.g. via single-crystal, directionally-solidified processing), pin the boundaries with particles to obstruct sliding, and lower the homologous temperature by choosing a higher-melting-point base metal.
The argument rests on a competition between two stresses that depend on strain rate in completely different ways.
The yield stress of a body-centred-cubic metal is dominated by the Peierls–Nabarro lattice friction acting on screw dislocations, whose motion requires the thermally activated nucleation of kink pairs. Anything that reduces the time available for thermal activation — a higher strain rate — or that reduces the thermal energy available — a lower temperature — raises the stress needed to move the dislocation. The result is a strongly rate- and temperature-dependent yield stress, conventionally written
$$\sigma_y \;=\; \sigma_y^{*}(T,\dot{\varepsilon}) + \sigma_G$$in which only the athermal component $\sigma_G$ (from solutes, precipitates and boundaries) is rate-independent.
The cleavage fracture stress $\sigma_f$, by contrast, is set by the stress required to propagate a sharp microcrack — a cracked carbide or a slip-band crack — through the surrounding matrix, essentially the Griffith condition $\sigma_f \approx \sqrt{2E\gamma_{\text{eff}}/\pi c}$. Neither the elastic modulus, the surface energy nor the microcrack size $c$ is significantly rate-dependent, so $\sigma_f$ is nearly flat with strain rate.
Plot the two on the same axes against strain rate and they cross. Below the crossover the material yields first, blunts any microcrack it produces, and fails in a ductile mode by void growth. Above it, the stress required to make the material flow has risen above the stress required to make a crack run, so the first microcrack formed at the yield point immediately propagates: fracture occurs at or before general yield, with almost no plastic work, and the fracture surface is cleavage. This is the same crossover that produces the ductile-to-brittle transition temperature (Question 4b), and it explains the well-known observation that increasing the strain rate raises the DBTT — a steel that is tough in a slow bend test can be brittle in a Charpy impact test at the same temperature.
Two amplifications complete the answer. First, a real structure concentrates the effect: at the tip of a sharp crack the local strain rate exceeds the nominal rate by orders of magnitude, and the triaxial constraint ahead of the tip raises the effective yield stress by a further factor of about three, so the crack tip is always the most brittle place in the component. Second, the contrast with rate-insensitive materials is a genuine structural distinction, not just a numerical one: in face-centred-cubic metals such as austenitic stainless steels, copper and aluminium, the Peierls stress is negligible, the yield stress is nearly athermal, and $\sigma_y$ therefore never overtakes $\sigma_f$ at any accessible rate or temperature. Those alloys show no ductile-to-brittle transition at all, which is exactly why austenitic stainless steel and aluminium are specified for cryogenic and impact-loaded service in preference to a ferritic steel of the same nominal strength.
