22-Mec-A4 Design and Manufacture of Machine Elements · December 2014

Question 2 of 6: Recovery and recrystallization of a strain-hardened metal

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examinations, December 2014 — 07-Mec-A4, Design and Manufacture of Machine Elements. Three hours, open book, any non-communicating calculator permitted. Six questions divided into Part A (Q1–Q3, manufacturing processes) and Part B (Q4–Q6, machine elements); candidates answer two from Part A and two from Part B, and all questions carry equal value (25 % each). All six questions are worked here.

Reference texts.

S. Kalpakjian and S. R. Schmid, Manufacturing Engineering and Technology — Ch. 10–12 (casting processes and design), Ch. 1 and 3 (structure, cold work, annealing), Ch. 21–22 (machining and tool life).
R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design — Ch. 3 (stress, Mohr's circle, beam deflection), Ch. 4 (statically indeterminate shafts), Ch. 16 (clutches and brakes).
R. C. Hibbeler, Mechanics of Materials — Ch. 6–9 (bending, transverse shear, combined loading, stress transformation).
W. D. Callister and D. G. Rethwisch, Materials Science and Engineering — Ch. 7 (dislocations, strengthening, recovery and recrystallization).

it reports a single 5,200 N load and a dimension chain of 750 + 450 + 450 = 1,650 mm that "contradicts" the 2,400 mm overall. Q5 is solved against the drawing, not the caption. Likewise in Q4 the 3,000 lb load is read from the drawing as a horizontal force applied through the bolted plate on the neutral axis, and the section as a 8 in deep I-beam (¹⁄₂ + 3¹⁄₂ + 3¹⁄₂ + ¹⁄₂).

Question 2: Recovery and recrystallization of a strain-hardened metal (25 marks)

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) The operative atomic mechanisms

Both processes are responses to the same stored driving force. Plastic deformation at low homologous temperature multiplies dislocations from an annealed density of order 10¹⁰ m⁻² to something approaching 10¹⁵ m⁻², and tangles them into cells. Perhaps one to five per cent of the plastic work done is retained in the lattice as the elastic strain energy of those dislocations; the remainder appears as heat. That retained energy makes the cold-worked state thermodynamically unstable, and on heating the metal spends it. How it spends it is what distinguishes the two processes.

Recovery proceeds by the motion and mutual annihilation of existing dislocations, without the creation of any new grains. At modestly elevated temperature vacancies become mobile enough to permit dislocation climb, and screw segments cross-slip. Two edge dislocations of opposite sign that lie on nearby slip planes can therefore come together and annihilate, removing both. Those that survive migrate into the lowest-energy arrangement available to them, which is a vertical wall of like-signed edge dislocations — a low-angle tilt boundary. This rearrangement is called polygonization, and it subdivides the original grain into subgrains misoriented from one another by a fraction of a degree up to a few degrees. Crucially, the original grains keep their identity, their shape and their crystallographic orientation: an elongated, cold-rolled grain remains an elongated grain. Point defects also anneal out, which is why electrical resistivity recovers almost fully during this stage, and macroscopic residual stresses relax. Because a substantial dislocation density survives, the strength is only slightly reduced.

Recrystallization proceeds instead by nucleation and growth of entirely new, strain-free grains. Small, nearly perfect volumes — typically subgrains that have grown, or regions at prior grain boundaries and deformation bands where the local misorientation is greatest — develop high-angle boundaries with their surroundings. A high-angle boundary is mobile in a way a low-angle boundary is not, and it sweeps through the deformed matrix. Every atom it passes is transferred from the strained lattice to the new, essentially dislocation-free one. The driving force is the difference in stored dislocation energy across the boundary, and the process consumes the cold-worked structure entirely, replacing elongated grains with new equiaxed ones. Dislocation density falls by four to five orders of magnitude, back to the annealed value. The property changes are therefore drastic rather than marginal.

Grain structure through cold work, recovery and recrystallization.

Three consequences follow that are worth stating explicitly, because they are what an examiner is testing. First, recrystallization requires a minimum amount of prior cold work; below a critical strain (of order a few per cent) there is insufficient stored energy and no driving force, and the metal will only recover. Second, the recrystallization temperature is not a material constant but falls as the prior strain rises and as the annealing time is extended, being roughly 0.3–0.5 of the absolute melting temperature. Third, recrystallization is the reason hot working does not work-harden: above the recrystallization temperature the structure re-forms as fast as it is deformed. If heating continues past recrystallization, the new grains coarsen by grain growth, driven now by grain-boundary area rather than by stored strain energy, which softens the metal further via the Hall–Petch relation and is normally undesirable.

(b) Effect on yield strength, tensile strength and total elongation

Effect of annealing temperature on strength (left) and total elongation (right).

The two diagrams show the characteristic behaviour. Through the recovery range, yield strength and tensile strength fall only slightly and elongation rises only slightly, because the dislocation population is rearranged but not substantially removed. Through the recrystallization range both strengths drop steeply and elongation rises steeply, as the strain-free grains replace the work-hardened matrix. Two details are worth noting. The yield strength falls proportionally further than the tensile strength, so the yield-to-tensile ratio drops and the metal recovers its capacity to strain-harden. And the elongation curve is very nearly the mirror image of the strength curves, which is the familiar strength–ductility trade-off read in reverse: annealing buys back the ductility that cold work spent.

Topic. Annealing of a cold-worked metal: recovery, recrystallization and grain growth, and the property changes each produces.

Key relations. Stored energy drives both stages. Strength scales with dislocation density through the Taylor relation $\tau = \tau_0 + \alpha G b \sqrt{\rho}$, so a fall of four orders of magnitude in $\rho$ cuts the strengthening contribution by a factor of about one hundred. After recrystallization the grain size governs strength through Hall–Petch, $\sigma_y = \sigma_0 + k_y d^{-1/2}$, which is why subsequent grain growth softens the metal further. Recrystallization kinetics follow a Johnson–Mehl–Avrami form, $X = 1 - \exp(-Bt^{\,n})$, and the recrystallization temperature falls with increasing prior strain.

Why this works. The distinction the question asks for is between rearranging defects and replacing the lattice. Recovery is a dislocation-level process: climb and cross-slip let dislocations annihilate in pairs and settle into low-angle walls, so the grain structure survives and only the properties that depend sensitively on point defects and long-range internal stress — resistivity, residual stress — change much. Recrystallization is a boundary-migration process: a high-angle boundary is a far more efficient sink, and once one is established it consumes the deformed volume wholesale. Because strength depends on the square root of dislocation density, only the wholesale process moves it appreciably, which is exactly what the two diagrams display.

Common pitfall. Treating recovery as "partial recrystallization" and drawing both curves as one smooth decline. They are mechanistically distinct and the property curves have a visible knee between them. A second frequent error is to draw the strength falling to zero; it falls to the fully annealed value, not to nothing. A third is to forget that recrystallization needs a critical prior strain, which is why a lightly worked part may never recrystallize no matter how long it is held.

Source. Callister & Rethwisch, Materials Science and Engineering, Ch. 7.12–7.13 (recovery, recrystallization, grain growth); Kalpakjian & Schmid, Manufacturing Engineering and Technology, Ch. 1.6 (cold, warm and hot working; annealing).

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