Question 5 of 7: Copper Wire Drawing — Penultimate Die Diameter for a Strength + Ductility Spec; Dislocations in Cold Rolling

Question 5: Copper Wire Drawing — Penultimate Die Diameter for a Strength + Ductility Spec; Dislocations in Cold Rolling (5 of 5)

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.

Given. Finished wire $d_f=0.7$ mm; UTS $>325$ MPa; ductility $\ge10\%$ EL; starting stock 10 mm rod, reduced by alternating cold draws and anneals; Fig 2 gives copper’s ultimate strength and elongation against %cold work.

Find. (a) The die diameter of the penultimate draw, i.e. the size the wire is drawn to and then annealed at, just before the final draw to 0.7 mm. (b) Dislocations and their role in cold rolling brass.

Approach

An anneal removes all previous cold work but does not change the diameter. So the finished properties depend only on the cold work of the final draw, from the penultimate (annealed) diameter down to 0.7 mm. Going straight from 10 mm to 0.7 mm would be $1-(0.7/10)^2=99.5\%$ CW, far off the chart. Steps: read the %CW window that meets both floors on Fig 2’s copper curves, choose a %CW inside it, then back out the penultimate diameter from $\%CW=\left[1-(d_f/d_p)^2\right]\times100$ (page-1 formula with $A\propto d^2$).

1. Strength floor → minimum %CW. On the printed copper ultimate-strength curve (digitised: $\approx211$ MPa at 0%, $285$ at 20%, $299$ at 25%, $324$ at 35%, $334$ MPa at 40% CW), the 325 MPa line is crossed at $$\%CW_{\min}\approx35.5\%.$$
2. Ductility floor → maximum %CW. On the copper elongation curve ($\approx40\%$ at 0%, $22\%$ at 20%, $14.9\%$ at 30%, $11.8\%$ at 35% CW), 10% EL is reached at $$\%CW_{\max}\approx38\%.$$ The window $35.5\%\le\%CW\le38\%$ exists but is narrow.
3. Choose the design cold work. The window is only about 3 points wide, so aim inside it rather than at an edge, so chart-reading and process scatter cannot push the wire out of spec: $$\boxed{\%CW_{\text{final draw}}=36\%}$$ Check against the curves: UTS $\approx326$ MPa $>325$ and EL $\approx11\%\ge10\%$.
4. Penultimate die diameter. $$\%CW=\left(1-\frac{d_f^2}{d_p^2}\right)\times100\ \Rightarrow\ d_p=\frac{d_f}{\sqrt{1-0.36}}=\frac{0.7}{0.800}$$ $$\boxed{d_p\approx0.875\ \text{mm}\ \ (\approx0.88\ \text{mm})}$$ The full window corresponds to $0.872\le d_p\le0.890$ mm. The wire is drawn to this size, annealed, and then given the final 36% cold draw to 0.7 mm. The earlier draw/anneal cycles from 10 mm do not affect the finished properties.
5. (b) What is a dislocation? A dislocation is a line-type crystal defect along which the regular atomic stacking is disrupted. In its two limiting forms: an edge dislocation is the edge of an extra half-plane of atoms inserted into the lattice; a screw dislocation is a helical distortion produced by a shear offset of one part of the crystal relative to another. The magnitude and direction of the lattice distortion a dislocation carries is captured by its Burgers vector $\mathbf b$. Dislocations move (glide) along specific close-packed crystallographic planes and directions (slip systems) under an applied shear stress; this glide motion is the physical mechanism of plastic (permanent) deformation in crystalline metals.
6. (b) Role of dislocations in cold-rolling brass. As the brass sheet ($\alpha$-brass, FCC, 12 $\{111\}\langle110\rangle$ slip systems) is rolled, plastic strain is accommodated entirely by dislocation glide on these slip systems. Every increment of plastic strain requires new dislocations to be generated (chiefly by Frank–Read sources operating within grains) and existing dislocations to move further, so the total dislocation density rises steeply with %CW — from $\approx10^6$ mm$^{-2}$ in the annealed sheet to $\approx10^9$–$10^{10}$ mm$^{-2}$ after heavy rolling. As density rises, dislocations increasingly intersect, tangle with one another, and pile up at grain boundaries and other dislocations on intersecting slip systems; each such obstacle impedes further dislocation motion, so a progressively larger applied stress is needed to keep deforming the material. This is exactly the mechanism of strain (work) hardening: the same dislocation-multiplication process that carries the plastic strain is also what raises the flow stress (and hence $\sigma_{UTS}$) and lowers the remaining ductility as %CW increases — precisely the TS-up/ductility-down trend captured in the Fig. 2 curves used in part (a).

[Figure not reproduced: Fig. Q5(a) — copper (solid) and iron (dashed) ultimate strength vs. %cold work, redrawn from the printed Fig 2. The copper curve reaches the 325 MPa floor at $\approx35.5\%$ CW. See the official exam paper.]

[Figure not reproduced: Fig. Q5(a) — copper (solid) and iron (dashed) elongation vs. %cold work, redrawn from the printed Fig 2. The copper curve falls to the 10% EL floor at $\approx38\%$ CW, so only the band $\approx35.5$–$38\%$ CW meets both requirements. See the official exam paper.]