4.2.2. Excessive Thinning of the Outer Diameter during Drawing

*ε* → → *ε* ൬ ට − ሺ ሻ ൰ The relationship between the total strain εtotal and the outer diameter during drawing *D*total was investigated. The deformation of the micro tube on the die's entrance side due to the back tension (state (0-2) shown in Figure 17) was small when compared to the main deformation (state (0-1)→(ii)). Therefore, this deformation was neglected. It is assumed that the outer diameter decreased during drawing from the state where the outer diameter matched with the die diameter *D*die, as (i)→(ii) shown in Figure 17. The outer diameter during drawing *D*total was calculated using the die diameter *D*die, the wall thickness after passing through the die approach *t*die, the Lankford value *r* in deformation during drawing, and the total strain εtotal. The wall thickness after passing through the die approach *t*die and the Lankford value *r* were calculated using Equations (3) and (4), respectively. The detailed deviations of Equations (3) and (4) are shown in the Appendix A.

$$t\_{\rm die} = \frac{1}{2} \left( D\_{\rm die} + \sqrt{D\_{\rm die} - \frac{4t\_0(d\_0 - t\_0)}{\beta}} \right) \tag{3}$$

$$r = \frac{\ln(D\_{\text{\tiny{}}}/D\_{\text{\tiny{\text{\tiny{}}}}})}{\ln(\,^{\iota\_{\text{\tiny{}}}/\iota\_{\text{\tiny{}}})}} \tag{4}$$

൫ ൗ ൯ ±ඨ ష ష Equation (5) indicates the total strain during drawing. By substituting Equation (4) into Equation (5) to eliminate the wall thickness *t*total, Equation (6) is obtained.

ሺtotalሻ

$$\varepsilon\_{\text{total}} = \ln \frac{l\_{\text{total}}}{l\_{\text{die}}} = \ln \frac{t\_{\text{die}}(D\_{\text{die}} - t\_{\text{die}})}{t\_{\text{total}}(D\_{\text{total}} - t\_{\text{total}})} \tag{5}$$

$$r = \frac{\ln(D\_{\text{total}}/D\_{\text{dis}})}{\ln \frac{D\_{\text{total}} \pm \sqrt{D\_{\text{total}} - \frac{D\_{\text{dis}} - t\_{\text{clle}}}{\exp\left(\varepsilon\_{\text{total}}\right)}}}} \tag{6}$$

ି n × The outer diameter during drawing *D*total was calculated using Equation (6). The excessive thinning of the outer diameter during drawing η' was calculated using Equation (7). Furthermore, the excessive thinning of the outer diameter after drawing η was calculated using Equation (8). The'measurement results of the final outer diameter *Dn* are already shown in Figure 8.

$$\eta' = \frac{D\_{\text{die}} - D\_{\text{total}}}{D\_{\text{die}}} \times 100 \tag{7}$$

$$\eta = \frac{D\_{\text{die}} - D\_n}{D\_{\text{die}}} \times 100\tag{8}$$

Figure 18a shows the theoretical relationship between the total strain εtotal and the excessive thinning of the outer diameter during drawing η'. η' increases as the total strain εtotal increases, and further grows during drawing as the Lankford value *r* increases. η' is greater than zero at any Lankford value. Therefore, it is considered that the outer diameter always becomes much smaller than the die diameter during drawing at any Lankford value *r*. Figure 18b shows η' in this study. Figure 18c–e show the Lankford value of each η' in Figure 18a. η' was obtained by substituting *t*die, εtotal, and *D*die into Equation (6) for each drawing condition. The theoretical excessive thinning of the outer diameter during drawing η' was larger in the order of the aluminum alloy, the copper, and the stainless-steel tube corresponding to the order of the Lankford values *r* in Figure 6. Therefore, the relationship of the magnitude between the theoretical values for each material seems to be appropriate, and the excessive thinning of the outer diameter during drawing η' increased as the Lankford value grew for each material.

*η* **Figure 18.** Theoretical relationship between the Lankford values *r* and the excessive thinning of the outer diameter during drawing η'. (**a**) The excessive thinning of the outer diameter at the Lankford values *r* of 0.01, 0.5, and 1.0. (**b**) The excessive thinning of the outer diameter in this study of all materials in this study, including (**c**) stainless-steel, (**d**) copper, and (**e**) aluminum alloy tubes. The dotted lines or curves in (**b**–**d**) indicate the eye guide.

#### Δ*ε* 4.2.3. Excessive Thinning of the Outer Diameter after Drawing

Δ*ε*

Δ*ε ε ε ε* = ( ି ) ( ି ) The relationship between the unloading strain ∆εunload and the final outer diameter *D<sup>n</sup>* (state(iii) in Figure 17a,b) was investigated. The unloading strain ∆εunload was calculated as the difference between the total strain εtotal and plastic strain εp, which is required for deformation to the final dimensions described by Equations (9) and (10).

$$\varepsilon\_{\rm P} = \ln \frac{l\_n}{l\_{\rm die}} = \ln \frac{t\_{\rm die}(D\_{\rm die} - t\_{\rm die})}{t\_n(D\_n - t\_n)} \tag{9}$$

*η η*

$$
\Delta \varepsilon\_{\text{unload}} = \varepsilon\_{\text{total}} - \varepsilon\_{\text{p}} \tag{10}
$$

Figure 19 shows the relationship between the unloading strain ∆εunload and the final excessive thinning of the outer diameter η. The final excessive thinning of the outer diameter η decreased when the unloading strain ∆εunload increased. Therefore, the final excessive thinning of the outer diameter decreased as the drawn tube recovered elastically during unloading. Δ*ε η η* Δ*ε*

ሺ ି ሻ ሺ ି ሻ

*ε ε*

*ε* =

Δ*ε η* **Figure 19.** Relationship between the unloading strain ∆εunload and the final excessive thinning of the outer diameter η, (**a**) all materials, (**b**) stainless-steel, (**c**) copper, and (**d**) aluminum alloy tubes. The dotted lines indicate the eye guide.

Δ*ε* Δ*ε* Δ*ε* The physical meaning of the unloading strain ∆εunload is discussed as follows. The elastic strain ∆εE, which depends on the elastic modulus of the bulk metal, was significantly smaller against the unloading strain of the loading-unloading tensile test, as shown in Figure 16. The elastic strain ∆ε<sup>E</sup> in hollow sinking was calculated using Equation (11).

$$
\Delta \varepsilon\_{\rm E} = \sigma \% \text{E} \tag{11}
$$

The parameter *E* is the elastic modulus of the bulk metal. The reference value of the elastic modulus of each bulk metal is substituted into Equation (11). The reference values of stainless-steel, copper, and aluminum alloy are 204 GPa [23], 119 GPa [23], and 69 GPa [26], respectively. The elastic strain value in the range of 0.0005 to 0.002 was significantly smaller against the unloading strain of 0.002 to 0.015. Therefore, the outer diameter of the micro tube approached the die diameter due to the unloading strain, which was larger than the elastic strain, during unloading. The excessive elastic

strain ∆εe, which was the difference between the unload strain ∆εunload and the elastic strain ∆εE, was calculated using Equation (12).

Δ*ε* Δ*ε* Δ*ε*

Δ*ε* Δ*ε* − Δ*ε*

$$
\Delta \varepsilon\_{\rm e} = \Delta \varepsilon\_{\rm unload} - \Delta \varepsilon\_{\rm E} \tag{12}
$$

Δ*ε* Δ*ε*

Figure 20a,b show the relationship between the excessive elastic strain ∆ε<sup>e</sup> and the apparent elastic modulus *E* ′ , and the drawing stress σ<sup>l</sup> . The excessive elastic strain ∆ε<sup>e</sup> grew when the apparent elastic modulus increased, or when the drawing stress decreased. The final outer diameter approached the die diameter as the drawing speed ratio decreased in Figure 8. Furthermore, the final outer diameter *Dn* fell in the order of the stainless-steel, the copper, and the aluminum alloy tube as shown in Figure 8. Figure 21 schematically shows the physical meaning of the excessive elastic strain. The micro tube recovered elastically to a degree greater than the elastic strain during unloading. Therefore, it is considered that the dislocations generated by microscopic yielding that occurred during drawing, disappeared partially during unloading. This phenomenon seems to be equivalent to the Bauschinger effect. It is considered that the dislocation density remaining after unloading was small under the conditions where few dislocations were generated during drawing such as with a high apparent elastic modulus, or low drawing stress. Therefore, the excessive elastic strain increased when the apparent elastic modulus grew, or the drawing stress decreased.

Δ*ε σ* **Figure 20.** Relationship between the excessive elastic strain ∆ε<sup>e</sup> and (**a**) the apparent elastic modulus *E* ′ , (**b**) drawing stress σ<sup>l</sup> . The dotted curve and line indicate the eye guide.

Δ*ε* Δ*ε σ σ σ* Δ*ε* **Figure 21.** Schematic illustration to explain the physical meaning of the excessive elastic strain ∆εe. The excessive elastic strain ∆ε<sup>e</sup> (**a**) at small drawing stress σ<sup>l</sup> and small apparent elastic modulus *E* ′ , (**b**) at large drawing stress σ<sup>l</sup> and small apparent elastic modulus *E* ′ , and (**c**) at large drawing stress σ<sup>l</sup> and large apparent elastic modulus *E* ′ . The parameters *E* and ∆ε<sup>E</sup> are the elastic modulus of the bulk metal and the elastic strain, respectively.

The unloading strain/total strain during the tensile test of the copper tube in Figure 16 was larger than that of the drawing experiment in Figure 19. Generally, the tensile residual stress is generated in the longitudinal direction during drawing [25]. Since the direction of the forces associated with the unloading behavior and the tensile residual stress are opposite, the tensile residual stress is considered to hinder the unloading behavior after drawing. Therefore, the amount of strain recovery during the tensile test was larger than that of the drawing test. A significant difference in tensile residual stress occurs when the die half angle is changed for the tube drawing process. Tube drawing was performed in this case using only one die half angle. Therefore, the effect of the tensile residual stress on the unloading strain is negligible. Investigation of the effect of the tensile residual stress on the unloading strain remains a subject of future research. Δ*ε* Δ*ε σ σ σ* Δ*ε*

An approximation of the unloading behavior was performed to investigate the effect of the apparent elastic modulus on the unloading behavior. Figure 22 shows the approximation method for the unloading behavior.

**Figure 22.** Illustration of the approximated unloading behavior.

 − 

> ሻ

> ሺ−

*ε ε*

*Metals* **2020**, *10*, 1315

The unloading behavior was approximated by a quadratic function, as shown in Equation (13). The parameter a is constant. By considering that the slope at the point (εtotal, *E* ′ ·εtotal) is *E*, Equation (14) is obtained.

$$
\sigma\_{\text{true}} = \left\| \mathbf{a} \right\|^2 \tag{13}
$$

$$\mathbf{a} = \frac{E}{2(\varepsilon\_{\text{total}} - \varepsilon\_{\text{p}})} \tag{14}$$

The plastic strain ε<sup>p</sup> is obtained by substituting εtotal and *E* ′ ·εtotal into ε and σtrue of Equation (15), respectively, as shown in Equation (15). *ε ε ε ε σ*

$$
\varepsilon\_{\rm p} = \varepsilon\_{\rm total} - \frac{2E' \cdot \varepsilon\_{\rm total}}{E} \tag{15}
$$

Lastly, substituting the plastic strain into Equation (15), Equation (16) is obtained as follows.

$$
\sigma\_{\text{true}} = \varepsilon\_{\text{total}} - \frac{E^2 \left(\varepsilon - \varepsilon\_{\text{total}} + \frac{2E' \cdot \varepsilon\_{\text{total}}}{E}\right)^2}{4E' \cdot \varepsilon\_{\text{total}}} \tag{16}
$$

Figure 23 compares the actual plastic strain calculated by Equation (9) and calculated by Equation (15). The experimental values of the total strain εtotal, the bulk elastic modulus *E*, and the apparent elastic modulus *E* ′ are substituted into Equation (15). The plastic strain obtained by Equation (15) generally agrees with the value obtained from Equation (9). Therefore, the unloading behavior can be expressed by a quadratic approximation. Figure 24 shows the unloading behavior obtained by Equation (16) at the bulk elastic modulus *E* and the apparent elastic modulus *E* ′ of 190 GPa and 95 GPa, respectively. The experimental value of the total strain εtotal is substituted into Equation (16). The strain completely recovered after unloading. Therefore, it is considered that the dislocations caused by the microscopic yielding disappear completely during unloading when the apparent elastic modulus is larger than a threshold value. *ε ε*

**Figure 23.** Comparison of the actual plastic strain and the value calculated using the approximate equation.

σ **Figure 24.** Unloading behavior when the true strain completely recovers. The parameter σ<sup>l</sup> is the drawing stress. The parameters *F*<sup>l</sup> and *A* are the drawing force and the cross-sectional area of the drawn tube, respectively. The parameters *E* and *E* ′ are the elastic modulus of the bulk metal and apparent elastic modulus, respectively.
