21-Mat-A5 Phase Transformations and Thermal Treatment · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2019 — 10-Met-A5, Mechanical Behaviour and Fracture of Materials. Three hours, closed book, Casio/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 crystallography of slip and twinning, dislocation theory, creep, fatigue, toughness and fracture mechanics, and safe-life fatigue design.
Q4(a) states explicitly that the plate is semi-infinite and gives a service tensile stress (450 MPa), so it is solved using the free-surface geometry correction that detail calls for, alongside a cross-check against the simpler through-crack assumption.
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 slip system is the combination of a specific crystallographic plane (the slip plane, the plane of highest atomic density) and a specific crystallographic direction lying in that plane (the slip direction, the direction of shortest atomic repeat distance) on which dislocation glide occurs. Both are set by the crystal structure: close-packed planes and close-packed directions minimise the Burgers vector and hence the energy of the glide dislocation.
In FCC the slip plane family is $\{111\}$ (four distinct planes) and the slip direction family is $\langle110\rangle$ (three independent directions lying in each plane), giving $4\times3=\mathbf{12}$ slip systems in total. Not all twelve are crystallographically independent, however: von Mises showed that an arbitrary shape change of a polycrystalline grain requires five independent slip systems, and of the twelve FCC systems only five are linearly independent (the other seven are expressible as combinations of those five, because the twelve systems are not all geometrically distinct strain contributors). FCC readily supplies the required five, which is the crystallographic reason FCC metals (Cu, Al, Ni, austenitic steel) are reliably ductile in polycrystalline form, in contrast to HCP metals (Question 4c).
A stereographic projection maps the orientation of every plane in a crystal onto a single flat diagram by projecting the plane's pole (its unit normal) from the far pole of a reference sphere onto the equatorial plane. For a cubic crystal, the standard $(100)$ projection is centred on the $[100]$ pole, with the primitive great-circle net (a Wulff net) providing angular coordinates, and the low-index poles $\{100\}$, $\{110\}$ and $\{111\}$ plotted at their known angular separations.
The key distinction the sketch must communicate is between the single-crystal projection on the left, which plots every symmetry-equivalent pole of one crystal at its exact geometric position, and a pole figure on the right, which plots one specific pole (say $\{110\}$) repeatedly, once per grain, for a whole polycrystalline sample. A random polycrystal produces a uniform density of points over the net; a textured polycrystal — produced by rolling, drawing or recrystallisation — produces intensity maxima at the orientations preferentially selected by the deformation or annealing history, exactly as sketched for the $\{110\}\langle112\rangle$ rolling texture common in BCC sheet steel.
A twinning system is, analogously to a slip system, a specific crystallographic plane (the twin/composition plane, across which the lattice is mirrored) together with a specific shear direction in that plane (the twinning shear direction) that together produce a homogeneous simple shear reorienting a thin lamella of the crystal into its mirror-twin orientation. Unlike slip, twinning shear is directional: shearing the correct sense produces the twin (a low-energy, mechanically favourable reorientation), while shearing the opposite sense along the same plane and direction does not reproduce a twin at all, so twin systems are not related by a simple $\pm$ sign the way slip systems are.
FCC twins on $\{111\}$ planes in the $\langle112\rangle$ shear direction. There are four distinct $\{111\}$ planes, and each contains three $\langle112\rangle$-type directions that lie in it, giving $4\times3=\mathbf{12}$ possible twinning systems — numerically the same count as the FCC slip systems, because the same $\{111\}\langle$in-plane$\rangle$ geometry underlies both, but twinning uses the $\langle112\rangle$ partial-dislocation shear direction rather than the $\langle110\rangle$ full-dislocation slip direction.
A twin trace is the line where a twin plane intersects a polished and etched surface; its orientation on the section depends only on the twin plane's orientation in space, not on the shear direction within it. The number of distinguishable trace orientations that can appear is therefore the number of crystallographically distinct planes in the twinning-plane family, counting a plane and its geometric opposite ($hkl$ and $\overline{hkl}$, which define the same infinite plane) only once.
So a heavily deformed FCC crystal can show at most 4 differently oriented twin traces on a given section, while a heavily deformed BCC crystal (which twins on $\{112\}\langle111\rangle$) can show at most 12. This higher multiplicity is consistent with BCC's greater propensity to twin under shock or low-temperature/high-rate loading, where more twin-plane variants compete to accommodate the imposed strain.
