4.2.1. Deformation Behavior of Micro Tube during Drawing and Unloading

*σ*

We have previously reported that the outer diameter of the micro tube decreased from the die approach end [10]. Therefore, the excessive thinning of the outer diameter was considered as a uniaxial

*β σ*

tensile deformation starting from the die approach end. To investigate this deformation behavior, the magnitude of the drawing stress in the stress-strain curve of the drawn tubes was evaluated. Figure 15a shows the stress-strain curve of the drawn stainless-steel tube at a die reduction *R*<sup>e</sup> of 0.29 and a drawing speed ratio β of 1.20. The dotted line indicates the drawing stress σ<sup>l</sup> during drawing. The state where the outer diameter matched the die diameter after passing through the die approach corresponds to the origin in Figure 15a. Generally, a bulk metal that has been deformed under a load in the elastic range will recover to its original shape as soon as the load is removed. In this study, the drawing stress σ<sup>l</sup> was in the macroscopic elastic region of the stress-strain curve. However, the outer diameter finally decreased excessively from the die approach end as the plastic strain. Therefore, it is considered that the excessive thinning of the outer diameter was caused by microscopic yielding of the micro tube due to the small number of crystal grains across the wall thickness, as shown in Figure 15b. Bulk metal, where many crystal grains exist across the thickness, yields microscopically even under macroscopically elastic deformation behavior [22]. Therefore, it is considered that the micro tube used in this study easily yielded microscopically because of the small number of crystal grains across the wall thickness. Furthermore, the apparent elastic modulus *E* ′ of the micro stainless-steel tube in Figure 15a was 24 GPa, which was much smaller than that of the reference value of bulk stainless-steel (204 GPa [23]). The reason for the decrease in the apparent elastic modulus due to the miniaturization reported elsewhere has not been clarified [24]. However, it is considered that this decrease in the apparent elastic modulus was also caused by microscopic yielding. The slope during loading in the macroscopic elastic range seems to decrease because of easy yielding for a certain drawing stress. The measurement accuracy of the true stress using the universal testing machine was confirmed in Section 3.5. Therefore, the apparent elastic modulus in Figure 5b seems to be appropriate.

*β σ* **Figure 15.** Measurement results of the stress-strain curve and drawing stress during drawing. (**a**) Stress-strain curve of the drawn stainless-steel tube at the die reduction *R*e of 0.29 and the drawing speed ratio β of 1.20. The dotted line indicates the drawing stress σ<sup>l</sup> . The parameters *F*<sup>l</sup> and *A* are the drawing tension and the cross-sectional area of the drawn tube, respectively. The parameter *E* ′ is the apparent elastic modulus of the stress-strain curve. (**b**) Schematic illustration of the microscopic yielding of the drawn micro tube during drawing.

*β ε* Δ*ε ε σ* Figure 16 shows the result of the loading-unloading tensile test of the drawn copper tube at a die reduction *R*<sup>e</sup> of 0.17 and a drawing speed ratio β of 1.10. Figure 16b shows the equivalent strain distribution during loading. The strain distribution between the chucks was uniform. Therefore, it is considered that the true strain can be evaluated by measuring the change in the chuck distance. The equivalent strain was also obtained using DIC by measuring the chuck distance. The total strain εtotal and the unloading strain ∆εunload obtained using the universal testing machine and the DIC matched with errors of 4.3% and 9.2%, respectively, as shown in Figure 16c. The measurement accuracy of the true strain using the universal testing machine was almost equivalent to that of the DIC in this

Δ*ε σ*

Δ*ε*

Δ*ε*

*ε*

Δ*ε*

*ε*

study. The measurement accuracy of the true stress using the universal testing machine was confirmed in Section 3.5. Therefore, the apparent elastic modulus in Figure 5b seems to be appropriate. Δ*ε* Δ*ε*

*β* Δ*ε* Δ*ε* **Figure 16.** Results of the loading-unloading tensile test of the drawn copper tube at the die reduction *R*<sup>e</sup> of 0.17 and the drawing speed ratio β of 1.10. (**a**) Loading-unloading curve. The parameters ∆ε<sup>E</sup> and ∆εtotal are the elastic strain and the total strain, respectively. The parameter *E* ′ is the apparent elastic modulus during loading. The symbols [i], [ii], and [iii] indicate the origin, the end point of loading, and the end point of unloading, respectively, (**b-1**) distribution of the equivalent strain, (**b-2**) magnified view of (**b-1**), and (**c**) comparison of the true strain obtained by the universal testing machine and the DIC. The symbols [i]–[iii] coresspond to Figure 16a.

The true stress εtrue increased during loading until reaching the drawing stress σ<sup>l</sup> , as shown in Figure 16a. The apparent elastic modulus during loading *E* ′ was 5 GPa, which was much smaller than that of the reference value of the bulk copper (119 GPa [23]). This decrease in the apparent elastic modulus due to miniaturization was also shown in the stress-strain curve of the starting material as shown in Figure 15. This decrease in the apparent elastic modulus was considred to be a result of microscopic yielding. The slope during loading seems to decrease because of easy yielding for a certain drawing stress. The true stress εtrue dropped slightly at a true strain of approximately 0.010 during loading. The reason for this stress drop is unclear. However, the true strain during this dropping was 0.7% against the total strain εtotal. Therefore, it is considered that this stress drop was negligible.

The true strain did not recover completely during unloading. The unloading strain/total strain was 0.31. This result also indicates that the micro tube yielded microscopically during loading. The outer diameter seems to approach the die diameter as the unloading strain ∆εunload increases. It is apparent that the unloading behavior depends on the elastic recovery outside the microscopically yielded region. Therefore, the unloading strain at the initial stage of the unloading behavior seems to depend on the elastic strain ∆εE, which was calculated by dividing the drawing stress σ<sup>l</sup> by the bulk elastic modulus *E* of 119 GPa [23]. However, the unloading strain ∆εunload was significantly larger than the elastic strain ∆εE. Therefore, it is considered that the outer diameter of the micro tube increased more than the linear elastic strain ∆ε<sup>E</sup> during unloading. The difference between the unloading strain and the elastic strain is defined as the excessive elastic strain ∆εe, which is discussed in Section 4.2.3.

According to the above discussion, the following two deformation behaviors should be investigated to clarify the mechanism that causes the excessive thinning of the outer diameter: (1) the excessive thinning of the outer diameter due to the microscopic yielding during drawing and (2) the unloading behavior caused by elastic recovery outside the microscopically yielded region during unloading, which determines the final outer diameter. Therefore, the followings were investigated: (1) the relationship between the total strain εtotal and the outer diameter during drawing *D*total and (2) the relationship between the unloading strain ∆εunload and the final outer diameter *Dn*. The detail procedures are shown in Figure 17. The outer diameter during drawing *D*total and the unloading strain ∆εunload could not be measured. Therefore, the outer diameter during drawing *D*total was calculated by using the Lankford value, which indicated the plastic anisotropy, and the total strain εtotal. The unloading strain ∆εunload was calculated by using the plastic strain ε<sup>p</sup> and the total strain εtotal. *ε* Δ*ε* Δ*ε ε*

The tensile residual stress, which is generated during drawing [25], seems to hinder the unloading behavior, because the force directions of the unloading behavior and the tensile residual stress are opposing. Tensile residual stress is not generated during uniaxial tensile deformation. Therefore, the difference in the unloading behavior between the tensile test and the drawing test is discussed based on the tensile residual stress in Section 4.2.3. Δ*ε ε ε*

**Figure 17.** *Cont.*


**Figure 17.** Road map for clarifying the mechanism causing the excessive thinning of the outer diameter. (**a**) Each strain in the stress-strain curve of the drawn tube. The parameters *E* ′ and *E* are the apparent elastic modulus and the elastic modulus of the bulk metal, respectively, (**b**) dimensions for each state. The symbol DD indicates the drawing direction. The parameters *F*<sup>l</sup> and *F*BT are the drawing tension and the back tension, respectively.
