*4.1. Conditions to Prevent Non Contact between the Tube and Die*

*β* We have previously reported that the copper tubes contacted the die under the conditions where the outer diameter on the die's entrance side was larger than the die diameter, as shown in the gray area of Figure 14b [10]. Therefore, the drawing speed ratio should be set low so that the die diameter increases, which prevents non contact between the tube and the die. Figure 14a,c show the outer diameter on the die's entrance side of stainless-steel and aluminum alloy drawn tubes, respectively. Similar to our previous report [10], the tube contacted the die at a small die diameter and a low drawing speed ratio. Only the aluminum tube contacted the die at a die diameter *D*die of 1.37 mm and a drawing speed ratio β of 1.50. In the following sections, the excessive thinning of the outer diameter was discussed only under the conditions where the tube contacted the die.

**Figure 14.** Outer diameter of the drawn tubes on the die entrance side for each material. (**a**) Stainless-steel, (**b**) copper, and (**c**) aluminum alloy tubes. The results of copper tubes were published in our previous paper [10].
