## 3 Experimental Investigations and Parameters Determination
### Anisotropic Yield Function and Hardening Parameters  
To employ the Barlat YId91 yield criterion for model predictions, both the elastic material properties and all six parameters in Equation (5) must be determined. To do so, the investigators performed a specific series of experiments, including tension and torsion loading, on specimens extracted at different orientations with respect to the longitudinal direction (Y direction as shown in Figure 2) from MB1 and MB2 bars. Details regarding the sample preparation and experimental set up are given in the Appendix A. Table 2 lists the detailed experimental plan for all specimens to determine Barlat YId91 parameters. The experimental setup for the LTD samples is shown in the Appendix A. True stress vs. true strain data for the uniaxial tension experiments (LDD, RDD0, RDD45, and RDD90 specimens) and shear stress vs. shear strain (\(\gamma\)) for the torsion experiment (LDT specimen) for MB1 and MB2 specimens are shown in Figures 3 and 4, respectively. The axial stress is calculated at each load level using the axial force and cross-sectional area,  
\[\sigma_{xx}=\frac{F_{x}}{A} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Rod Stock**} & \multicolumn{8}{c}{**Mechanical Properties**} & \multicolumn{8}{c}{**Chemical Composition**} \\ \hline \multirow{3}{*}{**Alloy**} & **Dia** & \multicolumn{3}{c}{**Ultimate Strength**} & \multicolumn{3}{c}{**Yield Strength**} & \multicolumn{3}{c}{**Elongation (\%)**} & \multirow{3}{*}{**Si**} & \multirow{3}{*}{**Fe**} & \multirow{3}{*}{**Cu**} & \multirow{3}{*}{**Mn**} & \multirow{3}{*}{**Mg**} & \multirow{3}{*}{**Cr**} & \multirow{3}{*}{**Zn**} & \multirow{3}{*}{**Ti**} \\  & **(mm)** & & & & & & & & & & & & & & & \\ \cline{3-19}  & **Min** & **Max** & **Min** & **Max** & **Min** & **Max** & & & & & & & & & & \\ \hline
6061-T6 MB1 & 28.575 & 317.2 & 327.5 & 286.1 & 299.4 & 16.5 & 18 & 0.71 & 0.28 & 0.33 & 0.05 & 0.89 & 0.05 & 0.02 & 0.02 \\ \hline
6061-T6 MB2 & 28.575 & 341.3 & 375.8 & 319.3 & 355.8 & 15.8 & 19.5 & 0.76 & 0.37 & 0.33 & 0.11 & 0.90 & 0.11 & 0.06 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties and chemical composition of Al6061-T6 MB1 and MB2 tubes given by the manufacturer (McMaster-Carr). Figure 1: Microstructure of (**a**) Al6061-T6 MB1, and (**b**) Al6061-T6 MB2 longitudinally extruded round bar specimens. Images obtained at 1000X using a Keyence microscope. The axes X and Y represent the radial and longitudinal directions, respectively. Figure 3: MB1 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. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Trial No** & **Mode** & **Load Cell** & **Strain Measurement Approach** & **Specimen** & **Material Al 6061-T6** & **Number of Experiments** \\ \hline
1 & Tension & MTS & Extensometer & LDD & MB1 & 2 \\ \hline  & & & & & MB2 & 2 \\ \hline  & & & & RD0 & & MB1 & 2 \\ \multirow{2}{*}{2} & \multirow{2}{*}{Tension} & \multirow{2}{*}{Psylotech micro- tensile tester} & \multirow{2}{*}{VIC 3D with stereo microscope} & \multirow{2}{*}{RDD45} & MB2 & 2 \\ \cline{4-5}  & & & & RD00 & & MB2 & 2 \\ \hline  & & & & RD00 & & MB1 & 2 \\ \multirow{2}{*}{3} & \multirow{2}{*}{Torsion} & \multirow{2}{*}{TestResources frame with torsion load cells} & \multirow{2}{*}{VIC 3D software with standard cameras} & \multirow{2}{*}{LDT} & MB2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental Scheme for determining material properties of Al 6061-T6 MB1 and MB2 bar. Figure 2: Coordinate system used throughout the developments. The \(X\)-axis is an arbitrarily selected radial direction that approximately corresponds to the bisector between the pair of stereo-cameras used for surface strain measurements (see Appendix A). The \(Y\)-axis is in the longitudinal direction.