21-Mat-A4 Deformation Behaviour and Properties of Materials · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, May 2017 — 12-Mtl-A4, Mechanical Behaviour and Fracture of Materials. Three hours, closed book, one approved non-communicating calculator. Eight questions of 20 marks each; the rubric marks only the first five questions as they appear in the answer book. All eight are solved here. This sitting's own printed content is fracture mechanics, fatigue crack growth, creep, toughness, strengthening, composites and plastic instability.
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.
(i) Creep requires a sustained (static, essentially time-independent) stress applied at an elevated homologous temperature, conventionally $T\gtrsim0.4\,T_m$ on the absolute scale. Under these conditions, thermally activated mechanisms (vacancy diffusion enabling dislocation climb, or grain-boundary diffusion/sliding) allow slow, continuous plastic strain accumulation even though the applied stress never reaches the athermal (room-temperature) yield stress. Time and temperature, not stress magnitude alone, are the controlling variables.
(ii) Fatigue requires a repeated (cyclic) stress or strain, applied many times, and can occur at ordinary (even sub-ambient) temperature. Each cycle produces a small amount of localized, reversible-looking slip at surface stress concentrators; damage accumulates irreversibly cycle by cycle (crack nucleation, then sub-critical growth) even though the peak stress in every individual cycle is well below $\sigma_y$ and would cause no visible damage if applied only once.
In the secondary (steady-state) creep regime, dislocations gliding on their slip planes encounter obstacles — precipitates, forest dislocations, solute atmospheres — that they cannot simply shear through or bow around at the applied sub-yield stress; at room temperature such a dislocation would simply stop (this is exactly why the athermal yield stress is not exceeded). At elevated temperature, however, vacancies generated (and diffusing) within the lattice can attach to or leave the dislocation core just below or above the obstacle, allowing the dislocation to "climb" a small distance out of its original glide plane and onto a parallel plane on which it can glide past the obstacle. Once past, it continues gliding until it meets the next obstacle and must climb again. Because this climb step is diffusion-controlled, its rate depends exponentially on temperature (through the self-diffusion coefficient, $D=D_0e^{-Q/RT}$) and is described empirically by Norton's power law $\dot\varepsilon=A\sigma^m\exp(-Q/RT)$, giving the essentially constant (steady-state) creep rate observed in stage II of the creep curve of Question 2(a). Because each individual climb-glide-climb cycle only needs to move the dislocation a fraction of the obstacle spacing at a time, the overall applied stress required is far below the athermal yield stress that would be needed to force the dislocation straight through the same obstacle field without any diffusional assistance.
Dislocation glide in some crystal structures (most notably BCC metals such as ferritic steel) is controlled by the thermally activated overcoming of the intrinsic lattice (Peierls) resistance, a process that is comparatively slow at the atomic scale; raising the applied strain rate forces dislocations to overcome this barrier with less time available for thermal assistance, so the yield stress required rises sharply with strain rate. The fracture stress (governed by fracture toughness and the pre-existing flaw population, $K=Y\sigma\sqrt{\pi a}$) is comparatively insensitive to strain rate by contrast. At a sufficiently high loading rate (impact loading), the rate-elevated yield stress can therefore rise above the largely rate-independent fracture stress before general yielding is reached at all — the material has no opportunity to relieve a local stress concentration by plastic flow, so it fractures in a brittle, low-energy manner instead of the ductile failure the same material would show under slow (quasi-static) loading. Materials whose yield stress is comparatively strain-rate insensitive (most FCC metals, e.g. austenitic stainless steel, copper, aluminum) keep their yield stress safely below the fracture stress across a much wider range of loading rates, so they retain ductile behaviour even under impact — which is exactly why BCC ferritic steels show a pronounced impact (Charpy) ductile-to-brittle transition while FCC metals generally do not.
