## 3 Experimental Investigations and Parameters Determination
### Anisotropic Yield Function and Hardening Parameters
The \(Z\) axis is another radial direction that is orthogonal to \(X\) and \(Y\). In these equations, \(F_{x}\) is the axial force, A = t \(\bullet\) w is the cross-sectional area of the rectangular specimens, where t is the specimen thickness and w is the width of the gage section,  
For the torsion experiments, the shear stress in the thin-walled tubular specimen is calculated for each applied torque using the torsion equation [62],  
\[\sigma_{xy}=\frac{T}{rA} \tag{10}\]  
where, \(T\) is applied torque, \(r\) is generally the mean radius for thin-walled specimens and \(A\) is the cross-sectional area given by \(A=\pi\left({{r_{\rm o}}^{2}-{r_{\rm i}}^{2}}\right)\), with outer tube radius, \(r_{\rm o}\), and inner tube radius, \(r_{\rm i}\). True stress (\(\sigma\)) and true strain (\(\epsilon\)) are calculated using standard formulae:  
\[\sigma=\sigma_{eng}\big{(}1+\varepsilon_{eng}\big{)} \tag{11}\]  
\[\varepsilon=ln\big{(}1+\varepsilon_{eng}\big{)} \tag{12}\]  
where \(\sigma_{eng}(\sigma_{xx})\) and \(\varepsilon_{eng}\) are engineering stress and engineering strain, respectively (Since strains are less than 0.05 in these studies, Lagrangian strain measurements obtained by StereoDIC are excellent estimates for the engineering strains, and hence are used in Equations (11)and (12) to determine true strain and true stress). The yield stresses in the three orthogonal orientations for MB1 and MB2 extrusions can be determined using the yield stress data for the LDD, RDD0, and RDD90 experiments. Three of the six anisotropic coefficients in Equation (5) are obtained from the three uniaxial stresses at yielding in the directions of the orthotropic symmetry axes X, Y and Z (Figures 2 and A1b) via a Newton-Raphson numerical procedure (The effective stress, (\(\sigma\)) from the LDD measurements for each bar material u considered to be the \"reference\" stress state, \(\sigma_{\rm o}\), and will be used to normalize results for both MB1 and MB2. As shown in Figure 3a, \(\sigma_{\rm o}\) = 287 Mpa for MB1 and Figure 4a, \(\sigma_{\rm o}\) = 342 Mpa for MB2). The effective stress, (\(\overline{\sigma}\)), from the LDD measurements is considered to be the \"reference\" stress state. The parameter, g, is obtained using (i) the estimated a, b, and c parameters, (ii) the plane stress transformation equations for the Z-X plane with \(\phi=-45^{\rm o}\) to relate the applied axial stress in the rotated specimen to the stresses in the X-Y-Z coordinate system, and (iii) Equation (5) to define an equation for g. Finally, to determine the parameters f and h, the data in Figures 3b and 4b for the RDD specimens shows that there is circumferential  
Figure 4: MB2 specimen results for (**a**) true stress vs. true strain measurements for tension loading of the LDD, RDD0, RDD45, and RDD90 specimens and (**b**) shear stress vs. shear strain for torsion of LDT specimens. symmetry in the longitudinally extruded rod material. Consistent with this observation, the authors assume that the yield stress due to shear are equal in both the X-Y plane and the Y-Z planes, so that f = h. Following this procedure, the six yield function parameters for the two 28.6 mm diameters extruded Al 6061 rods were determined, and the results are shown in Table 3 for both the MB1 and MB2 extruded aluminum bar materials. Figure 5 presents a comparison of the Barlat Yld91 anisotropic yield function predictions and the Von Mises isotropic yield function predictions for the case of biaxial loading in the X-Y plane. The procedure of obtaining Elastic and Power Law Hardening parameters is given in the Appendix A. Table 4 presents the results of elastic and power-law hardening parameters for both MB1 and MB2 bars.