The problem deals with the amount of cold work and its effect on material strength and ductility from mechanical properties of pure copper. A quantitative measure of amount of cold work is needed and one is provided in Chapter 2 in percent reduction in area:
This is also a logical measure of deformation imposed in drawing, and is the one used.
The cross-sectional area of a 3/8-inch rod is 0.110 square inch and the area of the 0.1 inch diameter product is 0.008 square inch. The total amount of cold work that would be performed in converting one to the other would be 93% - a value that would probably lead to wire fracture. Using figure 15-8, we find that final yield strength greater than 50,000 psi can be obtained with a cold work value greater than 27%. The desired ductility, greater than 10% elongation, can be achieved with cold work values less than 32%. Thus, if the final product can be produced with 30% cold work, both objectives can be achieved.
Therefore, the wire should be drawn from 3/8-inch diameter down to a determined diameter where it will be annealed to erase all cold work. Subsequent drawing from this diameter to the final 0.1 inch should then impart 30% cold work.
Calculate the anneal diameter using the equation,
Since the deformation from the starting condition of 3/8-inch diameter to the calculated value of 0.12 inch would probably exhaust the ductility of the wire and lead to fracture, additional intermediate anneals would probably be required. The maximum reduction possible in a wire drawing process is not discussed in the text. Quantitative models for maximum possible reduction exist. The general concept is that the drawing force acting on the wire should not result in a drawing stress that is larger than the yield strength of the wire leaving the die. Both the drawing stress and the wire yield strength depend on the amount of reduction, along with other factors such as die-work friction. Setting expressions for drawing stress and yield strength equal to each other gives the maximum reduction condition. By placing the last anneal at the diameter of 0.12 inch and deforming to the final diameter of 0.1 inch, the final product will be of both the desired size with the desired properties characteristics of 30% cold work.