**1. Introduction**

Micro metal tubes are used in various fields that entail medical devices and heat exchangers [1,2]. Thin tube walls are required to increase the volume flow rate of a liquid from needles during injection [3]. Low surface roughness of the tubes is also required to improve blood flow in injection needles when injecting liquid. Therefore, control over the wall thickness and outer surface quality are requirements of the micro tube fabrication process. A plug or mandrel is generally used to control the wall thickness

during tube drawing. However, inserting a plug or mandrel into micro metal tubes is impractical for creating micro metal tubes with a maximum outer diameter of 3 mm [4]. In conventional hollow sinking, where only the drawing speed on the die's exit side can be controlled, the wall thickness generally increases. Therefore, fabricating thin-walled tubes using conventional hollow sinking is difficult [5]. Tube volume entering the die is equal to that exiting the die over a single unit of time. Therefore, the final wall thickness decreases as the length of the drawn tubes increases. In other words, the final wall thickness decreases as the drawing speed ratio on the die's entrance and exit sides increases, when the outer diameter of the drawn tube matches the die diameter during the hollow sinking. This theory for controlling wall thickness has not been validated because only the drawing speed on the die's exit side can be controlled in a conventional drawing machine. However, a contemporary drawing machine that controls both the drawing speeds on the die's entrance and exit sides has recently been developed [6]. Previously, the authors investigated the conditions for wall thickness reduction by using this machine [7]. The results demonstrated that the wall thickness could be decreased when the drawing speed ratio was larger than a threshold value, which was obtained from the die reduction and the starting dimensions. Therefore, the drawing speed ratio must be set above this threshold to reduce the wall thickness during hollow sinking.

In our previous work, a theory for controlling wall thickness during hollow sinking was established [7]. Furthermore, a high dimension accuracy is required so that the outer diameter of the drawn micro tube matches the die diameter to improve the expansibility of stents [8]. In this study, tubes with a maximum outer diameter of 2 mm, which are required for several applications [5], were defined as the microscale tubes. Generally, the outer diameter of a macroscale drawn tube, such as an outer diameter of about 20 mm, matches the die diameter [9]. However, the authors reported that the outer diameter of the microscale drawn tubes with the outer diameter of about 1.5 mm became smaller than the die diameter starting from the die approach end [10]. Furthermore, the free surface roughening developed on the outer surface of the drawn tube due to the excessive thinning of the outer diameter. We assume that this excessive thinning of the outer diameter is caused by a uniaxial tensile deformation starting from the die approach end. In this study, we focused on the following aspects to clarify the mechanism causing the excessive thinning of the outer diameter: (1) flow stress, (2) friction force, and (3) plastic anisotropy. The details are described in the following paragraphs.

Several studies have reported that the size effect caused by miniaturization influences the deformation behavior of micro tubes [11,12]. For example, flow stress decreases with the grain number across the thickness of a specimen due to miniaturization [13]. The decrease in flow stress can be explained by using a surface layer model [14]. Dislocations pile up not at the free surface grain, but at the grain boundary. Therefore, dislocation movement in the free surface grains is less obstructed than movement at core grains. Furthermore, free surface grains exhibit lesser hardening than inner volume grains [15]. The fraction of grains representing the surface layer increases with miniaturization. Therefore, the flow stress decreases with the number of grains across the thickness. Several papers reported that this size effect appeared when only 10–20 grains exist in the thickness [11,12]. According to the above discussion, we assume that a microscale tube seems to yield more easily than a macroscale tube for a given drawing stress when the tube is stretched from the die approach end. The magnitude of the drawing stress during hollow sinking can be investigated using a finite element method (FEM) [16]. However, investigating complex phenomena such as the size effect by using this method is difficult. Therefore, the yielding behavior due to the size effect should be investigated by evaluating the magnitude of the measured drawing stress against the bulk yield stress.

The size effect on friction should also be considered when verifying the mechanism of the excessive thinning of the outer diameter during microscale hollow sinking. The force required to deform the tube decreases because of miniaturization even though the frictional force does not change. The proportion of the friction force to the drawing force increases with miniaturization. As a result, the micro tube seems to yield more easily during drawing than macroscale tubes because the drawing stress applied to the micro tube increases. The outer diameter of the micro tube seems to become smaller than the die diameter because of both the size effect on the flow stress and the effect of friction.

In addition to considering the size effect, investigating plastic anisotropy is expected to help clarify the mechanism causing the excessive thinning of the outer diameter during microscale hollow sinking. Several studies have reported on the effect of plastic anisotropy on the mechanical property of sheet metal forming by evaluating Lankford values [17]. The outer diameter of tubes decreases as the Lankford value increases in a tensile test. Therefore, the outer diameter is estimated to decrease from the die approach end as the Lankford value increases during hollow sinking. Generally, this value is calculated as the strain ratio of width to sheet thickness obtained via tensile testing. A study reported a method of measuring the Lankford value of tubes by realizing tensile testing on a specimen retaining a tube-shape [18]. The Lankford value of the tube can be calculated as the ratio of the circumferential strain ε<sup>θ</sup> to wall thickness strain ε<sup>t</sup> . From the above information, a more extreme reduction in the outer diameter due to an increase in the Lankford value should also be considered when verifying the mechanism causing the excessive thinning of the outer diameter during microscale hollow sinking.

We previously reported that the final outer diameter of the microscale copper tubes was smaller than the die diameter during hollow sinking [10]. The mechanism causing this excessive thinning of the outer diameter was not elucidated in our previous work. This mechanism will be clarified by considering the size effect and plastic anisotropy, which have not been considered in conventional macroscale hollow sinking.

This study aims to verify the mechanism causing the excessive thinning of the outer diameter during microscale hollow sinking. In this study, the stainless-steel, copper, and aluminum alloy tubes, which have been reported to have different Lankford values [18], were drawn to investigate the effect of the Lankford value on the outer diameter reduction during hollow sinking. The metals selected were adopted from the face-centered cube (FCC) metals so that the effect of the crystal structure on the deformation behavior could be neglected. A metal's crystal structure seems to affect the deformation behavior of the drawn micro tube. Furthermore, the micro tubes were drawn at several drawing speed ratios to confirm that they yielded easily due to miniaturization. The measurement results of the outer diameter and the outer surface roughness of copper drawn tubes were already reported in our previous paper [10]. However, additional experiments were conducted to investigate the deformation behavior of the micro tubes in more detail using the stainless-steel and the aluminum alloy tubes. Therefore, the contents in this study were deeper, more detailed, and original.
