## 5 Results and Discussion  
Effects of the tube material strength coefficient \(K\) on the deformation characteristic of the tube noosing process are analyzed by performing the FE simulation for five different values of \(K\); 500, 750, 1000, 1250, and 1500 MPa. The following process parameters are kept constant; \(E\) = 205 GPa, \(Y\) = 308 MPa, \(v\) = 0.33, \(n\) = 0.2, \(\mu\) = 0.1, \(D_{o}\) = 47 mm, \(L\) = 60 mm and \(t\) = 2 mm. Figure 4 shows the load-displacement curves during tube noosing for different values of the strength coefficients. It can be noticed that the location of the point of instability (point of the peak load) is not changed by the variation in the strength coefficient. This means that the limiting noosing ratio is not significantly affected by the variation in the strength coefficient of the tube material. Effects of work-hardening behavior for the used tube materials on the nosing load are analyzed by performing the FE simulation for five different values of the \(n\); 0.1, 0.2, 0.3, 0.4, and 0.5. The effects of the strain hardening behavior are carried out for the following constant process parameters: \(E\) = 205 GPa, \(Y\) = 308 MPa, \(v\) = 0.33, \(K\) = 1000, \(\mu\) = 0.1, \(D_{o}\) = 47 mm, \(L\) = 60 mm and \(t\) = 2 mm. Figure 5 shows the load-displacement curves during tube noosing for different values of the strain hardening exponent. It can be noticed that the die penetration increases with the decrease in the strain hardening exponent. An increase in the die penetration leads to further plastic deformation that can be achieved before the point of instability, i.e., a higher limiting noosing ratio can be reached. \begin{table}
\begin{tabular}{c c} \hline
**Properties** & **Values** \\ \hline Strength coefficient, \(K\) (MPa) & 500, 750, 1000, 1250, 1500 \\ \hline Strain hardening exponent, \(n\) & 0.1, 0.2, 0.3, 0.4, 0.5 \\ \hline Coefficient of friction, \(\mu\) & 0.05, 0.1, 0.15, 0.2, 0.25 \\ \hline Semi-die angle, (\(\alpha^{\circ}\)) & 10, 15, 20, 25 \\ \hline Tube outer diameter, \(D_{o}\) (mm) & 47 \\ \hline Tube wall length, \(L\) (mm) Tube & 60, 70, 80, 90, 100 \\ \hline wall thickness, \(t\) (mm) & 1, 2, 3, 4 \\ \hline [PERSON]’s modulus, \(E\) (Gpa) & 205 \\ \hline Poisson’s ratio, \(v\) & 0.33 \\ \hline \end{tabular}
\end{table} 
Table 2: Properties of the tubes and tooling conditions used for parametric study. Figure 4: Load–displacement curves for tube noosing with different strength coefficients for \(E\) = 205 GPa, \(Y\) = 308 MPa, \(v\) = 0.33, \(n\) = 0.2, \(\mu\) = 0.1, \(D_{o}\) = 47 mm, \(L\) = 60 mm and \(t\) = 2 mm. This result can also be seen in Figure 6, which represents the variations of theoretical, (Equation (13)) and FE values for limiting nosing ratio (LNR) with different values of the strain hardening exponent. This figure indicates that the limiting nosing ratio decreases with the increase in the values of the strain hardening exponent for the tube material. This is because a lower strain-hardening exponent has a lower yield stress in the deformation zone of the tube, which makes the tube nosing easily formed before it fails by buckling. The same tendency is reported in published work [16]. Effects of friction at the interface of the die and tube are analyzed by performing the FE simulation for five different values of the coefficient of friction, \(\mu=0.05\), \(0.1\), \(0.15\) and \(0.25\) The effects of friction on the deformation characteristic during tube nosing is carried out using the following process parameters \(E=205\) GPa, \(Y=332\) MPa, \(v=0.33\), \(D_{o}=47\) mm, \(L=60\) mm and \(t=2\) mm. The strain hardening behavior of the tube material is represented by  
\[\overline{\sigma}=867(0.00538+\bar{\varepsilon})^{0.2411}\]  
Figure 5: Load–displacement curves for tube nosing with different values of strain hardening exponent for \(E=205\) GPa, \(Y=308\) MPa, \(v=0.33\), \(K=1000\), \(\mu=0.1\), \(D_{o}=47\) mm, \(L=60\) mm and \(t=2\) mm.