Question 1 of 8: Cold-Drawn Copper Wire Design; Dislocations in Cold Rolling

Question 1: Cold-Drawn Copper Wire Design; Dislocations in Cold Rolling (20 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.

Given. Finished wire diameter $d_f=0.7$ mm; required UTS $>325$ MPa (46,000 psi); required ductility $\ge 10\%$ elongation; stock is 10 mm diameter rod, reduced to final size by an alternating sequence of cold draws and recrystallization anneals; Fig 1 gives copper's and iron's hardness, ultimate strength, and elongation as functions of %cold work.

Find. (a) The die diameter used for the penultimate draw — i.e. the diameter the wire is drawn to, then annealed, immediately before the final (ultimate) draw to 0.7 mm that sets the finished mechanical properties. (b) Definition of a dislocation and its role in cold rolling brass.

[Figure not reproduced: Fig. Q1a — Copper (solid) and iron (dashed) ultimate strength vs. %%cold work, redrawn from the source Fig 1. The copper curve crosses the 325 MPa requirement at %%CW ≈ 35%%. Curve values read from the printed Fig 1 (annealed copper 211 MPa at 0%%CW, 285&nbs. See the official exam paper.]

Approach

Because annealing removes all accumulated cold work but does not change the wire's diameter, the mechanical properties of the finished wire depend only on the %cold work accumulated since the last anneal — i.e. on the diameter the wire had immediately after that anneal, which is numerically the same diameter the penultimate draw produced (annealing changes microstructure, not size). The design procedure is therefore: (1) read Fig 1's copper curves to find the window of %CW that satisfies both the strength floor and the ductility floor for the final, un-annealed draw down to 0.7 mm; (2) pick a specific %CW inside that window; (3) back out the pre-final-draw diameter — which is the answer, the penultimate die diameter — from the %CW definition $\%\text{CW}=\left(1-(d_f/d_{die})^2\right)\times100$.

1. Read the strength-driven lower bound on %CW. On the copper ultimate-strength curve, UTS rises from ≈211 MPa at 0%CW (annealed copper), reaches ≈285 MPa at 20%CW, and only crosses the required 325 MPa line at $$\%\text{CW}_{\min}\ (\text{strength})\ \approx\ 35\%,$$ a chart reading, disclosed as approximate to the precision the printed curve supports (±2–3 percentage points).
2. Read the ductility-driven upper bound on %CW. On the copper elongation curve, %EL falls from ≈40% at 0%CW and reaches the 10% floor at $$\%\text{CW}_{\max}\ (\text{ductility})\ \approx\ 38\%.$$ Since $35\%<38\%$, a feasible window $\%\text{CW}\in[35\%,38\%]$ exists — the two requirements are simultaneously satisfiable, but only just: the strength floor and the ductility floor are barely three percentage points apart on this chart.
3. Select the design %CW. Because the window is narrow, the sound design choice is its midpoint rather than either edge, so that ordinary chart-reading and process scatter (±2–3 percentage points) cannot push the wire outside either specification: $$\boxed{\%\text{CW}_{design}=36\%}.$$ At 36%CW the copper curves read UTS ≈ 326 MPa (just above the 325 MPa floor) and %EL ≈ 11% (just above the 10% floor).
4. Solve for the penultimate die diameter. The %cold work definition (page-1 formula, with $A\propto d^2$ for a round wire) gives $$\%\text{CW}=\left(1-\frac{d_f^2}{d_{die}^2}\right)\times100 \ \Rightarrow\ d_{die}=\frac{d_f}{\sqrt{1-\%\text{CW}/100}} =\frac{0.7}{\sqrt{1-0.36}}=\frac{0.7}{0.800}$$ $$\boxed{d_{die}\approx0.88\ \text{mm}}$$ (equal, by construction, to the diameter the wire had immediately after its last anneal — annealing does not change diameter, only removes the work-hardened state).
5. (b) Dislocations in cold-rolled brass. A dislocation is a one-dimensional (line) crystal defect — an edge dislocation is the terminus of an extra half-plane of atoms inserted into the lattice; a screw dislocation is a helical distortion about a line, formed by a shear offset of one part of the crystal relative to another. Real dislocations are typically mixed, combining edge and screw character along their length. When a brass sheet is cold rolled, the compressive/shear stress imposed by the rolls resolves onto the crystal's close-packed slip planes and directions; dislocations glide along these planes, and plastic (permanent) shape change occurs one atomic-plane increment at a time as dislocations sweep through the grains and exit at grain boundaries or the free surface. Cold rolling also generates new dislocations (via sources such as Frank–Read sources) far faster than they can escape, so dislocation density rises by several orders of magnitude (from ≈$10^{6}$–$10^{8}$ to ≈$10^{9}$–$10^{10}$ mm of line per mm$^3$). The resulting dense, tangled dislocation network increasingly obstructs further dislocation motion (dislocations pin one another and pile up at grain boundaries and at the FCC-brass's characteristic annealing twins), which is the direct microstructural mechanism of strain (work) hardening — exactly the same cold-work strengthening read off Fig 1 in part (a).