## Q. 8.4

Design of a Wire-Drawing Process
Design a process to produce 0.20-in.-diameter copper wire. The mechanical properties of the copper to be considered are shown in Figure 8-6.

## Verified Solution

Wire drawing is the obvious manufacturing technique for this application. To produce the copper wire as efficiently as possible, we make the largest reduction in the diameter possible. Our design must ensure that the wire strain hardens sufficiently during drawing to prevent the drawn wire from breaking.
As an example calculation, let’s assume that the starting diameter of the
copper wire is 0.40 in. and that the wire is in the softest possible condition. The cold work is

%$CW = [\frac{A_{0} \ – \ A_{f}}{A_{0}}] × 100 =[\frac{(\pi /4) d_{0}^{2} \ – \ (\pi /4) d_{f}^{2}}{(\pi /4) d_{0}^{2}}] × 100$

= $[\frac{(0.40 \ in.)^2 \ – \ (0.20 \ in. )^2}{(0.40 \ in.)^2}] × 100 = 75$%

From Figure 8-6, the initial yield strength with 0% cold work is 22,000 psi. The final yield strength with 75% cold work is about 80,000 psi (with very little ductility). The draw force required to deform the initial wire is

$F = S_{y} A_{0} = (22,000 \ psi) (\pi /4) (0.40 \ in.)^2 =2765 \ lb$

The stress acting on the wire after passing through the die is

$S = \frac{F_{d}}{A_{f}}= \frac{2765 \ lb}{(\pi /4)(0.20 \ in.)^2}=88,000 \ psi$

The applied stress of 88,000 psi is greater than the 80,000 psi yield strength of the drawn wire. Therefore, the wire breaks because the % elongation is almost zero.
We can perform the same set of calculations for other initial diameters, with the results shown in Table 8-3 and Figure 8-13.
The graph shows that the draw stress exceeds the yield strength of the drawn wire when the original diameter is about 0.37 in. To produce the wire as efficiently as possible, the original diameter should be just under 0.37 in.

Table 8-3 Mechanical properties of copper wire (see Example 8-4)

 $d_{0}$ (in.) % CW Yield Strength of Drawn Wire (psi) Force (lb) Draw Stress on Drawn Wire (psi) 0.25 36 58,000 1080 34,380 0.30 56 70,000 1555 49,500 0.35 67 74,000 2117 67,380 0.40 75 80,000 2765 88,000